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014039PB 259 188·
Publication No. R75-34 Order No. 513
Seismic Design Decision Analysis
Report No. 21
PROBABILISTIC AND STATISTICALMODELS FOR
SEISMIC RISK ANALYSIS
by
Daniele Veneziano
July 1975
Sponsored by National Science Foundatiol1JResearch Applied to National Needs (RANN)
Grant 61-2795513
Additional copies may be obtained from
National Technical Information Service
. U.S. Department of Commerce
SEISMIC DESIGN DECISION ANALYSIS
Sponsored by National Science FoundationResearch Applied to National Needs (RANN)
Grant GI-27955X3
Report No, 21
PROBABILISTIC AND STATISTICAL MODELS
FOR SEISMIC RISK ANALYSIS
by
Daniele Veneziano
jUly 1975
Publication No. R75-34
Order No. 513
l\
1
Abstract
A number of models for engineering seismic risk analysis are proposed
and compared. In all cases, uncertainties are included both on the seismic
demand at the site, and on the seismic resistance of the facility. Particular
attention is given to the effects of inductive uncertainty on the model
parameters, which is due to limited available information. These parameters
include the mean occurrence rate of seismic events, the "decay rate" of the
frequency-site intensity law, the mean value and the variance of the resistance
distribution. The results from the models are compared with currently used
approximations, which are found to be unconservative. A numerical example is
presented, dealing with the estimation of seismic risk for nuclear power plants
located in Massachusetts.
2
Preface
This is the 21st in a series of reports under the general title ofSeismic Design Decision Analysis. The overall aim of the research isto develop data and procedures for balancing the increased cost of moreresistant construction agains the risk of losses during possible futureearthquakes. The research has been sponsored by the Earthquake EngineeringProgram of NSF-RANN under GrantGI-2785SX3. A list of previous reportsfollows this preface.
The analysis presented herein is oriented to the risk of failure(i.e inadequate preformenace) in a single, complex structure. A nuclearpower plant is used as an example - because of the work that already appears elsewhere in the literature concerning the behavior of such afacility. However, the theory applies equally well to important nonnuclear facilities.
Dr. Robert V. Whitman, Professor of civil Engineering is principalinvestigator for the overall research project, and the author is grateful to Professor R.V. Whitman for his encouragement in pursuing thiseffort and for his continuous helpful advice. Appreciation is also expressed for the critcal comments expressed by Professor C.A. Cornellon an earlier draft of this report.
3
List of Previous Reports
1. Whitman, R.V., C.A.Cornell, E.H.Vanmarcke, and J.W.Reed, "Methodology andInitial Damage Statistics," Department of Civil Engineering ResearchReport R72-l7, M.I.T., March 1972.
2. Leslie, S.K. and J.M.Biggs, "Earthquake Code Evolution and the Effect ofSeismic Design on the Cost of Buildings," Department of Civil EngineeringResearch Report R72-20, M.I.T., May 1972.
3. Anagnostopoulos, S.A., "Non-Linear Dynamic Response and Ductility Requirementsof Building Structures Subjected to Earthquakes," Department of CivilEngineering Research Report R72-54, M.I.T., September 1972.
4. Biggs, J.M. and. P.H.Grace, "Seismic Response of Buildings Designed by Codefor Different Earthquake Intensities," Department of Civil EngineeringResearch Report R73-7, M.I.T., January 1973.
5. Czarnecki, R.M.,"Earthquake Damage to Tall Buildings," Department of CivilEngineering Research Report R73-8, M.LT., January 1973.
6. Trudeau, P.J., "The Shear Wave Velocity of Boston Blue Clay," Department ofCivil Engineering Research Report R73-l2, M.I.T., February 1973.
7. Whitman, R.V.,S.Hong, and J.W.Reed, "Damage Statistics for High-Rise Buildingsin the Vicinity of the San Fernando Earthquake," Department of Civil EngineeringResearch Report R73-24, M.I.T., April 1973.
8. Whitman, R.V.,"Damage Probability Matrices for Prototype' Buildings,"Department of Civil Engineering Research Report R73-57, M.I.T., November 1973.
9. Whitman, R.V.,J.M.Biggs,J.Brennan lll,C.A.Cornell,R.de Neufville, and E.H.Vanmarcke, "Sunnnary of Methodology and Pilot Application," Department ofCivil Engineering Research Report R73-58, M.l.T., October 1973.
10. Whitman,R.V., J.M.Biggs, J. Brennan III, C.A.Cornell,R. de Neufville, and E.H.Vanmarcke,"Methodology and Pilot Application," Department of Civil EngineeringResearch Report R74-l5, July 1974.
11. Cornell,C.A. and H.A.Merz,"A Seismic Risk Analysis of Boston," Departmentof Civil Engineering Research Report R74-2, M.l.T., April 1974.
12. Isbell, J.E. and J.M.Biggs, "Inelastic Design of Building Frames to ResistEarthquakes," Department of Civil Engineering Research Report R74-36,M.I.T., May 1974.
13. Ackroyd, M.H. and J.M.Biggs,"The Formulation and Experimental Verificationof Mathematical Models for Predicting Dynamic Response of MultistoryBuildings," Department of Civil Engineering Research Report R74-37 MITMay 1974. ' •.• ,
4
14. Taleb-Agha,G.,"Sensitivity Analyses and Graphical Method for PreliminarySolutions," R14-4l, June 1974.
15. Panoussis, GO f "Seismic Reliability of Lifeline Networks," Department ofCivil Engineering Research Report R74-57, M.I.T., September 1974.
16. Whitman,R.V.,Yegian,M., J.Christian and Tezean, "Ground Motion AmplificationsStudies, Bursa, Turkey.
17. de Neufville, R.(Being written)
18. Tong,W-H, "Seismic Risk Analysis for Two-Sites Case," Department of CivilEngineering Research Report R75-23, Order No. 504, May 1975.
19. Wong, E.,"Correlation Between Earthquake Damage and Strong Ground Motion,"R75-24, Order No. 505, May 1975.
20. Munroe, T. and C. Blair,"Economic Impact in Seismic Design Decision Analysis:A Preliminary Investigation," Department of Civil Engineering ResearchReport R75-25, Order No. 506, June 1975.
BIBLIOGRAPHIC DATASHEET
. Title and Subtitle1
1. Report No.
MIT-CE-Rt7-34
152
3. Recipient's Accession No.
5. Report Date
SEISMIC DESIGN DECISION ANALYSIS
)robabalistic and Statistical Models for Seismic Risk Analysis
/. Author(s). .Danlele VeneZlano
J. Performing Organization Name and Address
Division of Constructed FacilitiesDepartment of Civil EngineeringMassachusetts Institute of TechnologyCambridge. Massachusetts 02139
12. Sponsoring Organization Name and Address
Office of Advanced Technology Applications, RANNNational Science FoundationWashington, D.C. 20550
15. Supplementary Notes
No. 21 in a series
16. Abstracts
July 19756.
8. Performing Organization Rept.No.
Order No o 51310. Project/Task/Work Unit No.
11. Contract/Grant No.
GI-27955X3
13. Type of Report & PeriodCovered
Task
14.
A number of models for engineering seismic risk analysis are proposed andcompared. In all cases, uncertainties are included both on the seismic demandat the site, and on the seismic resistance of the facility c, Particular attentionis given to the effects of inductive uncertainty on the model parameters,which is due to limited available informationo These parameters include themean occurrance rate of seismic events, the "decay rate" of the frequencysite intensity law, the mean value and the variance of the resistancedistributiono The results from the models are compared with currently usedapproximations, which are found to be unconservativ.e. A numerical example ispresented, dealing with the estimation of s.eismic risk for nuclear powerplants located in Massachusettso
17. Key Words and Document Analysis. 170. Descriptors
Reliability, Risk, Probability, StatisticsEngineering, Civil Engineering, Systems Analysis, System EngineeringEarthquakes, Nuclear Power Plants
l17b. Identifiers/Open-Ended Terms
Seismic Design Decision AnalysisSeismic RiskInductive Uncertainty
17c. COSATI Field/Group
18. Availability StatementRelease unlimited o
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5
Table of Contents
I. Introduction
II. Analysis of Uncertainty: Sources
.1 Uncertainty on the Seismic Demand
.2 Uncertainty on the Seismic Resistance
III. Analysis of Uncertainty: Types
IV. Probabilistic Seismic Damage Models
.1 Linear Gaussian Model
.2 Nonlinear Gaussian Models
(a) Truncated Linear AiN(b) Truncated Linear dA'~d.
~ ~N
(c) Truncated Linear d 2A' /d2 .~N ~N
(d) Quadratic Model
(e) Logarithmic Model
.3 Linear Ganuna ModelS
V. Statistical Seismic Damage Models
.1 Uncertainty on Demand Parameters
(a) Linear Gaussian Model; 1,.0 unknown, Sn known
(b) Linear Gaussian Model; 1,.0 known, Sn unknown
(c) Linear Gaussian Model; A and Sn unknown0
(d) Linear Ganuna Models
Page
7
10
10
18
40
44
44
46
46
48
49
50
51
52
67
67
67
68
68
70
.2 Uncertainty on Resistance Parameters 722
(a) lJR unknown, OR known 72
(b) 0 2R unknown, lJR known or unknown 73
.3 Uncertainty on Both Demand and Resistance Parameters 77
(a) 1 13 lJ k 02
k 77Aio ' I' R un nown; R nown
(b) A. , 13 1 , 0 2 unknown; lJR
known or unknown 79~o R
VI. Parameters Selection and Risk Evaluation 96
.1 Selection of Resistance Parameters
.2 Selection of Seismic Demand Parameters
.3 Mean Failure Rate Calculation
96
98
101
6
References
Most Commonly Used Symbols
Appendix A - Penalty Factors for Unknown OR and
Known or Unknown llRAppendix B - Penalty Factors for Unknown Demand
and Resistance Parameters
Page
107
112
114
127
7
1. Introduction
The seismic risk to which an engineering system is exposed depends on
two factors:
the future seismic demand at the site ("load"); and
• the (future) seismic capacity of the system ("resistance").
Description of the seismic load requires information at two different time
scales: at a macroscale, about the sequence of earthquake occurrences near
the site; at a microscale, about the detailed time history of the ground motion
for each future occurrence during the lifetime of the system. In a state of
uncertainty the earthquake sequence can be modeled- as a realization of a random
point process, and the indicidual microscale time histories as realizations of
continuous random processes.
The seismic capacity of the system can be described by a "resistance
vector" (assume a finite dimensional model), which collects the seismic response
characteristics and the performance criteria of the system as a whole, as well
as of its various subsystems and components. Some of the dynamic characteristics
may be time- or response-dependent; for example, the structural stiffness and
viscous damping.
The amount of information which is required for a complete probabilistic
description of both seismic demand and capacity is not within present knowledge
and analysis capability. Nevertheless, one can formulate simplified, yet
meaningful, models which demand much less information. In these simplified
models, the seismic load at the site is generally described by:
1. The mean earthquake occurrence rate, A, as a partial characterization
of the random point process. (Under the common assumption of Poisson
arrivals, A characterizes completely the occurrence process.)
2. The marginal probability distribution of a scalar (possibly vector)
"intensity parameter" Y, which replaces in approximation the
continuous random process model of the ground motion. Y might
measure (or include) the Modified Mercalli intensity at the site I,
the peak ground acceleration a, the peak ground velocity v, or any other
motion parameter which is correlated with the system's performance.
Also basic to these simplified models is the description of the seismic resistance
through a random damage function of intensity, D(Y), which accounts implicitly
for all the possible consequences of malfunctioning and failures of any part of
the system.
8
Clearly, there is a whole theory behind the quantification of local
seismicity parameters such as A and the distribution of Y (engineering
seismology); similarly, there is a whole theory behind stochastic damage
analysis (system reliability, random vibration), which can be used to calculate
the damage function D(Y).
The price for this simplified description of demand and capacity is that
seismic risk statements can only involve the mean rate of events producing
given damages (e.g., see Eq. 2). But this is not a critical limitation; although
neither the complete time characteristics of the damage process, nor the exact
nature of damage are given by the analysis, mean damage rates provide enough
information for the practical evaluation of seismic risk, and for comparison with
other natural threats. Clearly, if the occurrence of seismic events follows a
Poisson process and the resistance of the system does not depend on time, the
occurrence of damaging events is also a Poisson process. Under the weaker
condition that strong earthquakes occur as an approximately Poisson process
(which is a frequent assumption in engineering seismic risk analysis), the
probability of experiencing extensive damage during a period of time T is well
approximated by the expected number of such rare and highly damaging events in T.
Formally, the analysis of seismic risk proceeds as follows. Let
Fy(Y) = P{Y~lseisrnic event} be the cumulative distribution function (CDF) of
the site intensity measure whenever an earthquake occurs. Then the mean rate of
events with site intensity larger than y is:
A't = A [ 1- - f y ('})J . (1.1)
If FD!y(d!y) = P{D<dly=y} denotes the CDF of the damage caused by an earthquake
with site intensity y, the mean rate of events which cause damage in excess of d
is:
AD CeL)::::).. J [1- fD1y(etlt)] cLfy(~)' (1.2)
o.,ee ';f
In many cases it is not the whole function AD
(') which is of interest, but only
~D(df)' i.e. the mean rate of events damaging the system beyond a critical level
df. Such events are called "failures." Then the mean failure rate is, from
Equation (2):
Af = AD(cl f )= AJ Pf
(}) oLfyL~) , (1.3)
a.ce l'where Pf(Y) = l-bjy(df!Y) = probability of failure at intensity y.
This report is concerned with the seismic risk analysis of engineering systems
in the sense of Equation (3). Some attention is also given to conparing the
9
results from Equation (3) with past proposed approximations. One approximate
procedure, which has been used without due caution, is based on the following
reasoning. If ground motions more severe than the design earthquake (e.g., in
nuclear reactor design, the so-called Safe Shutdown Earthquake) occur with
mean rate ADES = A[l-FY(YDES)} , and if Pf(yDES ) is the probability of failure
at the design intensity, then the mean failure rate can be calculated as:
(L4)
Equation (4) yields unconservative (too small) estimates of Af for anYgi~
YDES • In fact,
Af = AJ P£ (~} cL F'j (J) 1- Af Ff ( J) . cL f y l ~) ~ A' Pf ( J'D E'») J uL f y ('t) ,aee!- Lj>~DES 'j> "DES
and the last expression equals the approximation ADES Pf(yDES ). The unconservatism
of using Equation (4) instead of (3) is quantified in the present study, and
correction factors are calculated, which depend on the seismic risk model and
model parameters.
Several of the assumptions made in this study are common to the technical
literature on engineering seismic risk, but the model as a whole is original. To
the author's knowledge, statistical uncertainty (although not new to seismic risk
formulations) was never extended to both damand and capacity parameters, and
indeed not even to capacity parameters alone. These extentions are conceptually
and sometimes numerically important. Finally, sensitivity analyses of Af in
Equation (3) of the type presented here were never reported.
The presentation is organized as follows. First, sources of uncertainty
(on the seismic demand and on the seismic resistance) and types of uncertainty
(deductive and inductive) are briefly reviewed; see Sections II and III. In
Section IV a few probabilistic models are studied, in which the functions Pf(Y)
and Fy(y) in Equation (3) are given analytical form. The effects of statistical
(inductive) uncertainty on the parameters of the probabilistic models are studied
analytically and numerically in Section V. Additional numerical results are
collected in the Appendices. Finally, Section VI discusses the choice of the
parameters for mean failure rate calculation, and presents some numerical examples.
10
II. Analysis of Uncertainty: Sources
As expressed by Equations (1.2) and (1.3\ seismic risk of engineering
facilities depends on the unknown seismic demand at the site (function Fy ) and
on the unknown seismic resistance of the system (functionSFDly and Pf ).
Information related to these functions is briefly reviewed here, and will be
used in Sections IV and V to construct probabilistic and statistical seismic
risk models. With regard to seismic demand, emphasis is on data and models
for Eastern U.S. regions.
11.1 Uncertainty on the Seismic Demand
A common assumption, which has obtained repeated validation from historical
records (Richter, 1958; Allen et aI, 1965; Esteva, 1968) is that in any given
region the instrumental magnitude M has exponential distribution:
(11.1)
This is a consequence of Richter's "linear" frequency-magnitude law, which
establishes that the log number of earthquakes exceeding magnitude m, 10glOnm,
decays linearly with m:
LOQ Tl. =: a. - bmoJ 10 rn (II.2)
a and b = S/ln 10 being regional constants.
Both from theoretical considerations (Rosenblueth, 1964; Rosenblueth and
Esteva, 1966) and from statistical evidence, it appears however that Equations
(1) and (2) have a limited magnitude range of validity. The upper limit, ml ,
varies from region to region, but in all cases is smaller than 9. If in addition
events of small size (say, with M<m ) are neglected, the distribution (1) assumes. 0
the doubly truncated exponential form (Cornell and Vanmarcke, 1969; Cornell, 1971):
o
,
,(II.3)
where ~ = [l_e-S(ml-mo)] -11
is a normalization constant.
11
Although the parameter 13 in equations (1) and (3) varies from region to
region, values reported from different parts of the United States show remarkable
consistency (see Table 1).
Other nonlinear frequency-magnitude relationships have been proposed. Among
others: the "bilinear law" (see, e.g., Esteva, 1974):
(11.4)
"quadratic law"
,,U1>m
, 111 ~ 1TI
{
ex, 1 ex p (- ~1 In)Pr {.M >:In} =
<X.2. ex'p (-132
m)
e (S2-Sl)m. )and a 2=al ' and the (here, truncated
ToksBz, 1970; Merz and Cornell, 1973):
where 132>131(Shlien and
{
1 , m ~ mo ,
5 -1
0
- K1111
• 11- e ~ (m - mo) 1- ~2 (J1l--1I1;)J (II. 5)Pr 111 >- In} =- L ' 1110 <. 111 <- m 1 ;
, ill ~m 1 ;
2 2Sl(ml -mo)+S2(ml -mo) -1
where Kml=[l-e ] (A condition on 131 ,132 and mo is clearly
needed to ensure that Equation 11.5 is an appropriate, i.e. non-increasing,
complementary CDF.)
Equations (4) and (5) generalize Richter's linear law (1); both have been
reported to fit well empirical complementary CDF's.
While the value of 13 is quite stable throughout the United States, there is
evidence of large regional variability in the upper bound magnitude mI. The
question of the upper size limitation is often discussed in terms of Modified
Mercalli (MM) epicentral intensity, since most of the historical data are
available in this form.
A number of relationships have been proposed between Richter's magnitude
and epicentral intensity 10 • Some of them, in the linear form
(II. 6)
are collected in Table 2. The parameters a i (a2 in particular) are quite stable
from region to region.
Due to the linearity of Equation (6), the frequency-epicentral intensity
law is of the same type as the assumed frequency-magnitude law. For example,
from the exponential magnitude distribution (3) and from Equation (6), it follows
that
12
.1- ~ 1
0•
, lo < i < i 1 (II. 7)
where i. (mj-al)/a2 j=O,l ,J
SIo azS,
Kil {l-exP[-SIo(il-io)]}-l
For typical values of (il-iO) and SIo ' Kil is very close to 1, and a good
approximation to equation (7) is:
{1._~I (i-io)e 0
o
. •,~ ~ 10 ,
, i o < 1 < i 1. ., 1. ~ II
(II. 8)
SIo can be estimated from Sand a2 if these parameters are known (see Table 1 and 2).
In other cases SIo ' or more generally the linear frequency-intensity law:
111 A'l
(Ai=mean annual rate of events with epicentral intensity in excess of i), have been
estimated directly from data on epicentral intensity. Table 3 collects some
proposed values for a o and SIo' The variability of SIo from region to region (or
from author to author) is explained in part by the subjective assessment of
epicentral intensities, and by the inclusion/exclusion of early, incomplete data.
As to the parameter a o ' it clearly depends on the seismic region and on its
extention. The estimates in Table 3 are therefore reported with the only purpose
of indicating typical values.
The question of whether an upper bound intensity i l ' or an upper bound
magnitude ml can be established with good confidence in a given region is quite
controversial. Upper bound magnitudes: ml =8.7 for the whole world, ml=8.5 for
California (Housner, 1970), ml=7.4 for Central United States (M & H Engineering,
1974) have been proposed. Housner (1970) has tentatively suggested the following
functional dependence of ml on the seismicity parameters in equation (2):
13
1
b, (II. 9)
where ml =8.S=magnitude upper bound for California;c
(ac ,bc)=(S.S,0.9)=seismicity parameters for California;
(a,b)=seismicity parameters for a generic region.
The upper bound epicentral intensities: il=lO for the New Madrid zone, i l =9
for the Matcog area (M&H Engineering, 1974), i l =6.3-8.7 for various sources in
the Boston area (Cornell and Merz, 1974), and the "maximum creditable" value
il=lO for the Mississippi Valley area (Howe and Mann, 1973) have also been
proposed"
In the Eastern United States, where regional seismicity is weakly correlated
with the known geological structure, and where bursts of activity often alternate
with periods of quiescence, the arguments against adopting moderate upper bounds
(say, i l =6-7) are rather convincing (Chinnery and Rogers, 1973; Howell, 1973;
Nuttli, 1974; Hausner, 1970). In Section V it will be shown that, depending on
the resistance characteristics of the system, the mean failure rate Af in Equation
(1.3) may not be sensitive to il. When applicable, this is a most welcome result,
due to the large uncertainty on and the open controversy about the upper bound
intensity.
For the purpose of seismic risk analysis one needs a measure of site
intensity. Throughout this study, such measure is taken to be either Modified
Mercalli intensity I, or alternatively, peak ground acceleration, a. Other motion
parameters, such as peak ground velocity or displacement could be used instead,
without altering the procedure, or the results to any significant degree.
A widely used relationship between site intensity I, epicentral intensity I ,oand epicentral or focal distance R is (see, e.g., Cornell, 1968):
I = C 1 -r C2. I () - C3 In R + t- , (11.10)
where Cl,CZ,C 3 are regional constants, and E is a random error term. For
the Northeastern United States, Cornell and Merz (1974) used a more general
attenuation law, of the form:
14
I, ;
(II.ll)
with parameters: Ro=lO miles
C ={2.6 for sites with rock foundations
1 3.1 for "average" soil conditions
C2=1. 0
C3=1. 3
The standard deviation of the zero-mean, normal error term € was estimated to
be about 0.2 for rock foundation sites and about 0.5 when including all possible
soil conditions at the site. The value C3=1.3 was found to agree quite losely
with data from Eastern United States regions.
Due to a higher absorbtion of wave evergy, in the western states intensity
attenuates much faster with distance (Algermissen, 1972; Brazee, 1972; Bollinger,
1973); for those regions a value of about 2.0 or 2.5 might be appropriate for
the coefficient C3 in Equation (11).
Given trle geometry of the active sources, their geographical location with
respect to the site, the mean rate of earthquake occurrences, the spatial
distribution of the epicenter, and the probability distribution of the epicentral
intensity for each source (the last distribution in the form, say, of Equation 8),
one can calculate the frequency-intensity law at the site through repeated appli
cation of Equation (11) (see Cornell, 1968, 1971; Cornell and Merz, 1974). For
a set of sources with no intensity upper bound, the exponential distribution of
epiceatral intensity:
,,
.,(11.12)
and a deterministic attenuation law (€=O in Equation 11), the complementary CDF
of the site intensity is:
Pr { I > i} ex (11.13)
IS
wherefor ~in<Ro'
R.mln= minimum distance of the site from the active sources,
Cl,io,aa= constants; same as in Equation (11).
Typical results are shown in Figure 1 (from Cornell" and Merz, 1974), where the
annual probability that Boston experiences an earthquake of intensity i or more is
plotted versus i. This probability is contributed by 8 separate seismic sources
located at variable distance from the city of Boston. Each curve corresponds
to a different set of parameters values, but in all cases it is Slo=l.lO and C2=1;
i.e., SI=l.lO. The upper curves, denoted UB12 and RANDOM 12, are for the case
of an upper bound epicentral intensity i l =12 for all sources. The slope of these
curves is almost identical with that of equation (13), with SI=l.lO (see the line
between dots in Figure 1). The increase of negative slope at high intensities
is due to the upper bound on 10
, The randomness of the attenuation law (a
standard deviation 0£=0.2 was used in Equation 11) has no appreciable effect on
the slope of the risk curve. In obtaining curve CA 12 it was assumed that the
upper bound epicentral intensity was 12 for two sources, and was variable in the
range 6.3-7.3 for the other 6 sources. The remaining curves result from smaller
upper bounds on la, this reduction causing a rapid risk drop at smaller levels of
site intensity.
Within the intensity range shown, the risk curves in Figure 1 are well
approximated either by straight lines with an "effective" slope parameter
SI=l.l (curves UB12 and RANDOM 12) or SI=1.77 (curve CA 12), or by truncated
straight lines with an effective SI between 1.70 and 2.00 (remaining curves).
These and other "nonlinear" seismic risk models will be studied in Section IV.
Result8 of a similar kind are shown in Figure 2 (from Liu and Dougherty,1975)
for a site at variable distance from the San Andreas fault. For the calculation
of the site intensity risk curve, the magnitude distribution (3) was used, with
parameters mo=4.S, ml=oo, S=0.87 In 10, and a mean occurrence rate over the
entire fault length (644 Km) of 6.33 events/year. The attenuation law, expressed
in terms of magnitude and focal distance was taken to be:
16
lei + C.2 11 - c 3 In R
8.1.6 + 1.4'5 .M - 2..461u R
Again, a linear relationship between log10 risk and I, with slope -b/C2=-0.6
(slope of the line between dots in Figure 2) provides a good approximation to
the risk curves for all but very small site intensities.
A parameter which is often used as a measure of seismic demand at the site
is peak ground acceleration. Empirical relationships have been established among
a, M and R, and between a and I, so that seismic risk curves in terms of a can
be evaluated (in approximation) either from knoWn frequency-magnitude relations,
or from site intensity risk curves. According to the best information presently
available (Esteva, 1970,1974; Esteva and Villaverde, 1973; Donovan, 1973,1974;
see also Newmark, 1974) the model
a. (11.14)
is in satisfactory agreement with the empirical data if, for a in g's and
L(R)=a linear function of focal distance in KID, the parameters b i are given the
values in Table 4. (Formally identical relationships have been suggested for
peak ground velocity.)
Several proposed relations between log acceleration and MM intensity are
shown in Figures 3 and 4. The solid line in Figure 4 used the most extensive
set of data.
Due to the approximate linearity of 1n a in M (equation 14) and in I
(Figures 3 and 4), the considerations about the exponential decay of the site
intensity distribution (Equation 13 and related comments) hold also for 1n a,
after replacing C2 by b2• For example, for a set of seismic sources with
magnitude distribution (1), the probability that the peak ground acceleration
a is exceeded during anyone event is proportional to exp{-ln a·S/b 2}.
In this study, both seismic demand and seismic resistance are characterized
in terms of MM intensity. For design,however, it is desirable to measure intensity
through actual characteristics of the motion, such as peak ground acceleration. The
relationships sketched in Figures 3 and 4 were fit to very dispersed data (see,e.g.,
Newmark,1974,and Ambraseys,1974). How to account for this dispersion when passing
from MMI to 1n a is not clear: simply "adding" it to the variability of MM
intensity (say, in the attenuation law) generates very large 1n a uncertainties.
17
What is more important, is that there are ways to calculate risk in terms of
peak ground acceleration which are more efficient, in the sense of producing
less dispersed results. One such way is to first convert epicentral intensities
into magnitudes (the empirical relationships in Table 2 show little dispersion;
see, e.g., Chinnery and Rogers, 1973), and then use an attenuation law giving
acceleration as a (random)_ function of magnitude and distance. The plots in
Figures 3 and 4 should therefore be regarded as best estimates of In a given
MM site intensity, not as functional relationships, and caution should be
exercised in their use.
18
11.2 Uncertainty on the Seismic Resistance
It is generally believed that the uncertainty in the seismic resistance
of engineering facilities contributes marginally to the overall risk (Ferry Borges,
1956; Rosenblueth, 1964; Vanmarcke and Cornell, 1969), and that even a seismic
risk model with deterministic resistance produces valuable results. The present
study reaches different conclusions, particularly when the analysis includes
statistical uncertainties. It appears, in fact, that a substantial fraction of
total risk may come from moderate intensity earthquakes which, although
associated individually with small failure probabilities, are much more frequent
than large and statistically more destructive events.
Three different approaches have been pursued to estimate the probability
distribution of system damage (this includes the probability of "failure," if
failure is defined as a particular damage state) for given seismic intensity:
(a) random vibration theory; (b) simulation of artificial ground motions and
repeated deterministic analysis of the system's response; (c) direct analysis of
damage statistics from past earthquakes. The main advantages and limitation of
each approach are:
(a) Random vibration analysis generally requires simple models, both
of the ground motion (e.g., a pseudo-stationary Gaussian process)
and of the system (e.g., linear elastic, with known parameters).
Apart from these limitations, random vibration techniques are most
powerful, in that they characterize the system's response as a
random process, from which the probabilities of reaching various
damage states can be calculated (approximately); see, e.g., Vanmarcke,
(1969) and Cornell (1971). Unfortunately, most structural systems
become highly nonlinear near collapse, or even after moderate damage.
In addition, if the size of the earthquake is known in terms of MM
intensity or of peak acceleration, it is not easy to relate these
parameters to a random ground motion process.
(b) Simulation methods (see, among others, Housner and Jennings, 1965; Hou,
1968) do not impose such strict limitations on the input and system
models; however, by their very nature, they generate information on
low probability events at prohibitive computational costs. Simulation
methods also become impractical if the behavior of the system is itself
uncertain.
19
(c) In recent years, much has been learned from the analysis of
actual damage statistics; although data on severe damage probabilities
are still scarce for some categories of buildings, information is
becoming available at an (unfortunately) high rate. Lack of
statistically relevant data is indeed the major limitation of
this approach. Advantages over (a) and (b) are that no assumption
is made on the seismic load or on the system behavior, and that
direct correlations are obtained between intensity parameters (say,
I or a), and damage.
In this study, the damage-statistics approach (c) is followed, with consideration
both of the estimated damage probabilities, and of the uncertainty on such
estimates due to limited data processing. Information and models of seismic
damage are reviewed in the remainder of this section.
Much information can be found in recent literature on the Mean Damage
Ratio (MDR=expected repair cost over total property value) for various categories
of buildings, exposed to ground motions of given intensity. Mean damage ratio
functions (of MMI) have also been fitted to the data, or estimated subjectively.
Although the damage statistics for some building categories (such as wooden
frame and masonry constructions) are of less direct interest to this study,they
are also reviewed briefly, since they provide further insight into the general
dependence of seismic damage on intensity and on seismic design.
Figure 5 (adapted from Mann, 1974) suwnarizes the information available on
wooden frame dwellings. The solid curve was proposed by Steinbrugge, McClure and
Snow (1969), as a result of a very extensive effort which combined field data,
past experience and subjective judgement. The damage values suggested by
Friedman and Roy (1969) are also judgemental; they were estimated by extrapolating
data on dwellings' damage from the 1957 San Francisco earthquake, the 1952 Kern
County earthquake and the 1933 Long Beach earthquake. These data are not strictly
comparable with the remaining data points in Figure 5, since they make no
distinction between types of dwelling construction (e.g., frame versus brick), or
existance of chimney. For wooden frame dwellings and in the intensity range 5
to 8, the MDR varies by a factor of approximately 4 per unit of intensity.
Data for ordinary and for reinforced masonry construction are summarized
in Figure 6 (from Mann, 1974). In this case the MDR for a given intensity is
20
very sensitive to the quality of construction and to the use of reinforcement or
not. Apart from the rapid decay of the expected damage at low intensity levels,
the dependence of MDR on I is approximately exponential (as for wooden frame
construction), now with a factor of about 3 per unit of intensity for ordinary
masonry, and of about 2.75 per unit of intensity for reinforced masonry. The
MDR for weak masonry is from 3 to 10 times the MDR for high quality masonry,
depending on the ground motion intensity. At high intensity levels, reinforcement
has the effect of reducing the mean damage ratio by a factor or approximately 5.
Damage statistics for high-rise buildings with steel-framed, concrete-framed,
and concrete-shear-wall structural systems have become available only in the very
recent past. Reliable information was collected after the 1971 San Fernando
earthquake (Steinbrugge et aI, 1971; Whitman et aI, 1973 a,b; Whitman, 1973).
The most extensive of these surveys (Whitman, 1973) documented 368 buildings
with 5 stories or more, calssified by age, by structural material, and by
height. Most of these buildings experienced a motion of intensity 7. At that
intensity, old (pre-1933) buildings, designed under no seismic requirement,
experienced a MDR about 2% greater than recent (post-1947) construction, designed
for the Uniform Building Code seismic zone 3 (UBC 3). On the average, steel
frame buildings performed better than concrete-structured buildings. Figure 7
(from Whitman, 1973) displays MDR data for high-rise buildings from the San
Fernando as well as from other earthquakes. While data are differentiated by
DBC zone, all heights and all types of construction (steel and concrete) are
lumped together. The data denoted "Japan" are from the 1968 Higashi-Matsuyama
and from the 1968 Tokachi-Oki earthquakes and refer to buildings designed for
lateral forces about 2 to 3 times greater than those for DBC zone 3.
Based in part on these empirical data, curves relating the MDR of high-rise
buildings to MM intensity have been proposed by several authors. Figure 8
(from Whitman, 1973) shows mean damage ratio functions evaluated subjectively
(by S.B.Barnes and Associates, Los Angeles) for l3-story concrete frame
buildings designed in compliance with various DBC zones, and for a "Superzone"
S, with twice the lateral force required for zone 3. Similar subjective
estimates have been made for other structural systems. Figure 9 (also from
Whitman, 1973) compares estimates for Concrete Shear Wall (CSW), Concrete
Moment-Resisting Frame (CMF), Steel Moment-Resisting Frame (SMF) , and Steel
Braced Frame (SBF) structural systems. By combining these subjective estimates
with empirical data on high-rise buildings, Whitman (1973) suggested the mean
damage ratios shown in Figure 10 (solid lines) as applicable to the population
of constructions mentioned above.
21
For high-rise buildings (5-stories or more) in Los Angeles, Whitman and
Hong (1973) proposed the dashed lines in Figure 10 (the dotted continuations
are extrapolations beyond the available data).
In the analysis of data from the 1971 San Fernando earthquake, Benjamin
(1974) found no statistically significant difference between the mean damage ratios
of high-rise reinforced concrete and steel constructions. He also observed that
log MDR is approximately linear in MMI, and suggested the straight lines (a)
and (b) in Figure 10 as probable bounds to the actual log MDR-I relationship.
The degree of correlation between "aseismic" design provisions and effective
damage protection is rather controversial. In some cases (see, e.g., McMahon,
1974, for damage to high-rise buildings during the 1972 Managua earthquake; and
Pique, 1975, for damage statistics from the 1974 Lima earthquake), comparable
mean damage ratios were found for buildings designed for different UBC zones.
However, the probability of high damage and collapse were notably and consistently
reduced by seismic protection, particularly in shear-wall constructions.
At the other extreme, cases were reported (e.g., Hong and Reed, 1972, on
the 1965 Puget Sound, Washington, earthquake) where aseismic protection was
apparently very effective. The same conclusions were arrived at by Mann (1974),
who compared the performance of skeleton framed buildings designed for UBC zones
o and 3, during various earthquakes (see Figure 11, where Class A refers to steel
and Class B to reinforced concrete constructions).
Evident, but not so extreme, beneficial effects of aseismic design were
found by Whitman (1973) for high-rise buildings (see Figures 8 and 10), and by
Crumlish and Wirth (1967) for school buildings in California and in Washington.
In all cases, as Whitman (1973) suggested, greater benefits are expected
in stiff buildings, if the increased design lateral force does not impare severely
the ductility of the system, and if the seismic resistances of various portions
of the structure are comparable. Similarly~ Newmark (1974) pointed out that
construction details, selection of materials, placement of reinforcement and
of stiffeners, quality control of welds and connections, more than the general
compliance with aseismic provisions are essential to reach high ductility factors
and therefore to resist strong ground motions.
Some information is available also on the conditional CDF of damage, FnlI(d/i),
i.e. the function which is used in Equation (1.2).
Benjamin (1974) found that for broad classes of buildings (rane;ing from
wooden frame dwellings, to light industrial constructions, to high-rise buildi.ngs)
22
the damage data for given intensity fit well both lognormal and gamma distributions.
If a lognormal model is used. then (log MDR!I) has normal distribution. Benjamin
also found that the variance of (Log MDR!I) is approximately constant with I.
For light industrial buildings he estimated:
0 Log MDRI I 0.295
= 0.225
for model (a) in Figure 10;
for model (b) in Figure 10.
The damage statistics reported by Whitman (1973) also indicate that 0Log
MDRII
is not sensitive to I; the same statistics are consistent with a normal
distribution of (Log MDR!I).
As indicated previously. the log mean damage ratio varies almost linearly
with the MM intensity. For the developments in Sections IV and V it is not the
absolute value of 0 Log MDRII which has importance. but the ratio
(3D -0-Lo~ .MDR \ 1.
, (11.15)
where bD is the slope of the linear relationship:
(11.16)
Table 5 collects some statistics and some subjective evaluations of the parameters
bD and BD. In a strict sense. the values of bD and BD from Newmark (1974) and
Vanmarcke (1971) cannot be compared with those from Benjamin (1974) and Whitman
(1973), because they refer to given peak ground acceleration a. instead of MMI.
Newmark suggested values of 01 ( )1 for ordinary buildings and forn response a
nuclear reactor structures and equipment (parameter BETA in his Table 3). If
the level of response is proportional to a. and a varies by a factor 2 per unit
of MMI as suggested by Figure 4 (but see earlier comments on Figures 3 and 4). the
response varies also by a factor 2 per unit of intensity; so that one can
estimate BD in equation (15) as:
In 2.
cr11'1 (response)la.
23
This relationship was used to calculate the Sn values in Table 5 from Newmark's
estimates of'01 ( )1 .n response a
The estimates of bn and Sn from Vanmarcke (1971) were found as follows. If
E[log MDRII] is linear in I (see Equation 16) and if 1n a has functional relation
ship with I (again, see Figure 4 and related comments):
'In a = I . 111. 2 - 7. 3 (11.17)
then E[log MDRla] is linear in 1n a, say:
whence:
It is also:
- a. + b in a.D,a D,a..
:::: a.n + In 2. . b . ID,a.
b . 111 2D, a
crLog M DR \ In a.
(11.18)
(11.19)
where 1n a is given by Equation (17). Given bn, a and 0Log MDRl1n a - estimates
of these parameters can be obtained from the data in Vanmarcke (1971)
bn and Bn can be calculated from Equations (18), (19) and (15).
A critical question is how all this information on the damage statistics and
on the resistance distribution of ordinary buildings relates to the behavior of
special constructions or of new structural typologies. The problem arises, for
example, in the seismic risk analysis of nuclear power plants, to which the
following considerations are primarily addressed. Statistical data on seismic
damage to nuclear power facilities are practically missing, so that the procedure
of extracting information from historical records no longer applies. Also the
analytical approaches (say, of the random v:Lbration type), which were found some
what inaccurate for damage prediction of oniinary buildings (Whitman, 1973) ,
encounter major difficulties here, due to the complexity of nuclear reactor
systems, to the sequentia1ity of accidental events leading to "failures," to
the built-in redundancy, and to the different levels of resistance of various
subsystems and components.
24
Nevertheless, some general conclusions can be drawn on the seismic
frequency of specific initiating events. In fact, for each initiating event
a single subsystem or component is involved directly, and some damage
characteristics of ordinary structures can be assumed to hold, at least qualitat
tively (e.g., the approximate linearity of the expected log "damage" as a
function of MMI). From Table 5, a range of values for Bn in Equation (15) can
be established (the values from Newmark were suggested specifically for nuclear
reactor structures and equipment). The question remains to be answered, what
is a reasonable value for the expected subsystem or component damage at a given
MMI (this would determine the parameter an in Equation 16), and what damage level
df should be associated with "accident initiation." In the context of the risk
model introduced in Section IV, the last two questions reduce to a single
question; for example, what is the seismic intensity at which there is 50% change
of accident initiation? Newmark (1974) estimated that at the design value of
peak ground acceleration the ratio
In (response at failure) - E[ln(respOhSe)]
cr111 (res' POl1 S'e)
for nuclear power plant structures and equipment exceeds by about 0.63 and 0.66,
respectively, the same ratio for ord1nary buildings designed for UBC zone 3.
This indication will be used in Section VI to relate the seismic risk of ordinary
buildings to the seismic risk of reactor structures and equipment.
.SEISMIC REGION
* Southern New England(Chinnery and Rogers,1973)
* New Jersey(Isacks and Oliver, 1964)
* Central Mississippi RiverValley(Nutt1i, 1974)
* North and Central America(Sh1ien and ToksBz, 1970)
* Southern California(Albee and Smith, 1967)
* California(Housner, 1970)
25
S COMMENTS
2.19(±0.12) 1800-1959; 135 events
2.17
2.00(±0.25) 1833-1972; 250,000 Km2
2.26 1963-1968
1.94 1934-1963; 10,126 events;296,000 Km2
2.07
r---------------------~--------- ----------------+-------------------------1
* Various Parts of the World(Evernden, 1970; Esteva,1968; Ferry Borges andCastaheta, 1971)
* World(Gutenberg and Richter,194l)(Hausner, 1970)
1.61-2.88
2.302.07 1904-1946
Table II.I-
Values af S in Equations (11.1) and (11.3) for
Different Seismic Regions
26
SEISMIC REGION
* Southern California(Gutenberg and Richter,1956)
* Southern New England(Chinnery and Rogers, 1973)
* Eastern United States shallow eqs.(Howell, 1973)
* Washington and Oregon(Algerroissen, 1969)
*(A1gerroissen et aI, 1969)
Table 11.2
M=l+~I3 0
M = 1. 2 + O. 6 10
M = 1.3 + 0.6 10
M = 0.82 + 0.69 10
M = 1.14 + 0.62 10
Proposed Relationships Between
Magnitude and Epicentral Intensity
27
SEISMIC REGION
* Southern New England(same for Boston area, southernNew Hampshire and Hartford area;Chinnery and Rogers, 1973)
* Southeastern United States (SouthernAppalachian, Central Virginia andSouth Carolina-Georgia zones;Bollinger, 1973)
6.93
1. 31 (±O. 07)
1. 36
* Northeast United States 1928-1967(Cornell and Merz, 1974)
* Bostor. area 1630-1970(Cornell and Merz, 1974)
* New Madrid Zone 1870-1970(M&H Engineering, 1974)
* Matcog Area 1870-1970(M&H Engineering, 1974)
* Mississippi Valley(McClain and Myers, 1970)
* Mississippi Va11ey-St.Lawrence(A1germissen, 1969)
* Central United States(Liu and Fage1, 1972)
* California(A1germissen, 1969)
* World 1534-1974(Cornell and Merz, 1974)
Table II. 3
1.05
3.62 1.10
7.64 1.43
5.02 1.34
3.41 0.93
6.24 1.17
4.49 1.15
9.03 1. 24
1.35
Parameters of the Linear Frequency-Epicentra1 Intensity Law:
In Aj = cx'O-BIe!Ai = mean annual rate of events in the entire seismic region with
epicentral intensity in excess of i
28
bl b2 b3
Esteva (1970) 1. 26 0.80 2.00
Donovan(1973) 1.35 0.58 1.52
Donovan (1974) 1.10 0.50 1.32
Table 11.4 Coefficients b i in the acceleration
magnitude-focal distance relation (11.14)
bD(**) BD
Benjamin (1974) model (a) in Fig.lO 0.484 1. 64model (b) in Fig.lO 0.347 1.54
Whitman (1973) Post-1947 BuildingsSan Fernando,I=6 ~ 1.13 z 1.88San Fernando,I=7 "" 1.13 <::: 2.13San Fernando,I=7.5 <;::: 0.91 "'" 1. 90
Newmark (1974)(*)Nuclear Reactor
Structure 1. 33Nuclear Reactor
Equipment 1.16Vanmarcke(1971)(*)
I~6. 7 ::::= 0.68 = 1.241-::::.7.8 ~ 0.59 ~ 1.53
Table U.S Values of bD and BD in equations (11.15) and (11.16)
(*)These values were not obtained from equations (11.1S) and (11.16);see explanation in the text.
(* *) .More than oneE(log MDR/l].the indicated
value of bD is given for proposed nonlinear functionsThe values correspond to local linearization around
MM intensity.
>-!:::(/)zW
CURVE~
~ AL
C) 163
z(5ww0xW
0::0
C) -4z 10-I<:[::>0w~
0
~(/)
~ 10-I<:[::>z~
29
Mean Return Period
.J.QQ..yeol"S _
1000 years
\--+--+--H-+----+-- .-- .---
\-6
10 y=----'-----:Jl[::::::----L-----=:m~--------L...-----'---JZllI~----.i----,lX-l....-----l..-.L-.JX
(FIRM GROUND) M.M. INTENSITY IN BOSTON
Figure II.I
Influence of Different Assumptions on the Seismic Risk
In Boston
(from Cornell and Merz, 1974)
1
30
10-2....1
;r<r""fH\I)
ZI.<.lH ~;ITE-TO-FAULTz -3H 10~
DISTANCEHH 10 Km\I)
<.j 25 flUzHq
10- 4;,..j 50 Kro~uX 100 KmI.<.l
~.t;
0
;r<HH -;...:l 10 ~Hr:!l<:lOG0~P;
I II III IV v VI VII VI II IX x XI XII
MODIFIED HLRCALLl INTENSITY, I
Figure 11.2
Probability of Exceeding Site.Intensi~
When an Earthquake Occurs Along the San Andreas Fault
(from Liu and Dougherty, 1975)
31
1000-.,.-------------------------n,..-----;rr------,EARTHQUAKE INTENSITY-ACCELERATION
RELATIONSHIPS
1_ 0.1,)
.0
100
100
"l>'"II> 1_0.01,)...... 10
:lE
'"Z •Z0
;::ota:.....J...'"'"<l 1.0
0.'
t n m = = Dt x
EQUIVALENT Ill,. INTENSITY
A- HERSHBERGER (1956)
B- GUTENBERG 6 RICHTER (lg42)
• C- CANCANI (1904)
·O-ISHIMOTO (lg32)
• [-SAVARENSKY a KIRNOS (1955)
*F-MEDVEOEV ET AL. (1963)'
*G.-N.I. DRAFT BY- LAW
H-T10-7024 (1963)
*I-KAWASUMI (1951)
*"-PETERSCHMITT (1951)
,lOATA FROM G.A. EIBY (1965)
Figure 11.3 Relationships between ~lli Intensity and Peak Ground
Acceleration
(from Linehan, 1970)
WN
XII
1. H&H Eng. (1974)X r 2. Gutenberg and Richter(1956)
3. Whitman (1971)4. Newmark (1974)
IX r---S. Richter;see M&H (1974)6. Housner;see M&H (1944)7. Ambraseys (1974)
VIn r M'%.....-·h'" ./ 7",,' -~' .. /
H VII~
:>-<HH
~ VIwHZH
~ V
~ I XX
IV r XX .... / 1'/ ' ~ a~1·1n 2 - 7.297x ...x ........2 X X, ......... " :yx '
III I ...1 , ..
... .-....~~
II
IO.OOlg a.Glg O.lg 1. Og
Peak Ground Acceleration, a
Figure 11.4 Proposed Relationships Between MMI an~ Peak Ground Acceleration.
Curve 7 used European Dat~
33
•
0
~0~0/'
0'7J 1-
/ -_/
/~
iI:;{
,.
1°
II
II
II
0;j
I Q Son Fernando 5 'V Long Beach 8
Santo Barbara4 Steinbrugge I -I 0 0
I x Sante Ros.:l 7 to Chorlesfon 4
F-riec:L.mah d11d -I II Roy (:1 969).
II0
7.0%
10/~
0.2%
0,5%
')
4
c:rl 3~-....~(Y~
~CJ:$ 2<:=l
;.-!;<i-Li~<
l~/
If VI VII VIII IX X
M.!'; site intensity, I
Figure II~S Hean Damage Ratio versus Intensity for
Wooden Frame Dwel!ings
(from Mann, 1974)
The solid curve is after Steinbrugge et a1 (1969).
34
INTENSITY I (MM)
VI6
VII7
V III8
IX9
X10
Cc
CcCd
*1 *2Il - --- ~ -~-1(Q_ __ _ _.J(l __..,
~- +7- 1./ "fJ. ~/ Aj,,'-' /0..., I
<0(,0 V / "" 'fl'
Oy-.~ ...rl'
V' v71 R-co 'J~. ~. & ',0" ..~ ' -0' ,(, + /~~,'l.,,--~<;,~_l. .I (\ 0IO?{ -a--.-:,'-;'------- ,.c;,;--- - - ~ I~--------
O~-~-., .. \"-----------. ,_,"1- -----------~ ~ --- _ ---.-- ----- -2;:§~-'---------
o,~, ~,V,y v-~-- . 0' .V0<:: <:J.<":-.c. __
,J ~..
1 k /I·~~l'Yo t/ +~.,o
~
4 ~ I, "~.10( .- ,.
°t-<t0:::
REFERENCE EARTHQUAKES
o Insuronance Underwriter
• Richter 98"V Long Beach - Martel
10'V LorYJ Beach - Hod9son (Schools)
b Santo BOiboro- Freerrci'l 4
+ Ker ns Co. - Stei nbr ugge - Mor an 6
X Santo Rosa - SteinbruQge 7
• Son Fernando -Steinbrugge 5
¢ Charleston _Freemon 4
*1 Occasional Collapse's
*2 Collapse 01 weaker structures and seriousof better masonry
eel Types of masonry Os described by Richter.C'!J
Figure 11.6 Mean Damage Ratio versus Intensity for
Ordinary and Reinforced Masonry Constructions
(after Mann, 1974)
10G.-.----,
Ap.it
, Ii i 'I! +r'" C:,C", I,'-\ ~ '- I
wV1
X
~o
100I
I
------1 -.L _
, Iii80 ~-'I-----'-'r----'1------"'r""-~"---1,--f----.-1----,---, --t--------r---,---
I Ii.. 1= i ' ! :Q 60--1,-----r"'-'---------+-----------'T, ---l
~ i I I i~ -~------t ' 1------+------(.9 i I' ~ :
:;! r=1 I~ 40 -1--------..--J----- --.- -- ----to I I I
I I I;z I I~ --1-----------1' --- ------~- -------- r----~ ~ I ' I II I I
20 ------t--- -- ---- ~---- I +---------: IiiIi..nt-~n~t---
°v Vi VII Viii IXxIX
SFe Sen Fernando
SF San Ffcncisco IC Cc;-ccosll. An:::ilorqeP Pugzt Sound
J Japan i j
Viii
40SFe, !
: l','~r----'-'.J --- --------! I! i
i
,, I
! ' 'i
~-,---------~---- '---'-'1i +C I
, I
i;1'.~r-e,p+c
VII
.~
ASh!i
.-JVI
S::-••SFe
.5;=-"
DES!C\~
.. USC 0
• USC 3
• lJ;?C 2
• J.cd"'AN
STR,4TC:GY
Cl ')' I ~."~I IV
Inv
~<:>
-0f-«N
';,.tJ,:...r:,....,.~,
2:.<.:"f.c)
~'-
d.,_I) o l i,.)'.'-
' II
MOO;FIED MERCALU INTENSITY MODIFIED MERCALlI INTENSITY
Figure 11.7 Data on the Mean Damage Ratio for High-Rise
Buildings
(after Whitman, 1973)
W0'
20
"
i 'IOOl i I I II I I II I 1I i I
I I I I• 80 1 ~ ~j
0: I' Io I~ I I I I Ja::: 60 I I Iw I
~ I~.:,':
;~ I ' i I
~ t I I /1r..! 4°1 'I 1- T Ild I I I" I
~ I I I ~0~",'I .' 1.-,,'"
• I ~// "1-
{T -.;/// I S.....'1
l- ' I~':'/: ~~ 1
I/'1,/'/"."..,~ --
o L../',' :-;.::, .-'" l ]V VI VII VIII IX
MODIFIED MERCALLI INTENSITY
~"'r'
t..'._
-:.."".,..-"•••~<~
<tG
L1J"'''';:<L.
~
o
iJJ<.::><
Vi VII Viii IX
MODiFIED MERCALLI INTENSITY
!~:.~
1----1----1 ---'-----I, 'I I II I I 'I ,I '! I Ii ! i• I i I I \lSC q,\
<' "I I I ~•• ~;;,..--! ...., l---- --- I L ---:>I - • ""I i - .-:,./ s
! 1 11-:"7-./11~I
, I /J./' II
I • II~/I /'//1 iI • 1 1/ ! IIl)';r;' L i !, I I
i I /,
i ;)'V! .' I (I
CU ,~ ! /'11 ,II--/--T v----.
VI .. If-i' - i '
, I, / I
l 1IIV!i
Figure 11.8 Subjective Mean Danlage Ratios
for High-Rise Concrete Frame Buildings (after Whitman t l973)
w"
~ ,I rt,,"I ,?, oX ~t---~c..'\'~i " .......... r~l.~' ",~
/' ,..~c\~r
USC 3
DESIGN STRATEGY
( b)
roo r 1
;;;.~
("":"1-.......ilJo:r~-;-...::
801 I I I 1
II I
rfl! , I ~
ei I I I I I::c 601 II I.~ I
t5 I ' ,~, I Io ~,(), I I
I I II I !
I I lI I I1 J I~ I I
I !
1-.-
c,"__
-:0:W",",::
iX
.o
!t.,]
~()
lJ/)Ir---- I !i I, i !I !'J') USC 0 II . .
I DES:G:~ STRATEGY ~ I 1. II i II I I
~"iiit.') ~----.-------T--------i I I
i I I II ! I .I 1 •
~ i II I: I
,..... II !r;'Y !------- k- ~
I
II
j i I
I, . I I,
g L---------1 1 ~ _
I 'I ","C/«I i ."/h,:".'
i I I I /h v "j, I , "/-t------i-- I ,,# ~ii' ,"'<:./ I
I I I " '0':'/~ . ' , '/ II -1" ; I
20r---- j 1,,'- V' \! . i "'-' /'1l.---------+-------- ,-:fZ--/' I ;
I ~ ..d/'''''' I~. ,.~- II ~ •
o ~ Vl- Vll VHI IX
MODiFIED MERCALLI INTENSITY
Figure 11.9 Subjective Mean Damage Ratios for High-Rise Buildings with Different Structural Systems
(after Whitman, 1973)
w00
x
I-1
~
!XVi VII VIII
MCDWIED MERCALLI INTEN'SITY°v
~o
100,-----· I I
1-+·····_-+···_·······1·---..80~-----+---- i - I-~--------J--'--------I-
I i II I Il-----·l. +._-----1-
~' I I : -
Q I I i 0'r- I i~ 60------r------t------Hgw I I I ~s ,'-------t.: ':,rt?: I 1« ~ io 401 I , , , I
Z r-----j----T------. i J
l5 I ! I
"r 11---201 -{ . t----I II .
~------1 l--i I I
IV."\/1 1! .Vii
Ii-------t--------------r----
I :, I
:·.,n2C)1F~ED ~,~'lERCA:_Ll lNTE~JS~TY
\/1
_._.._._._--~_._-_._-+._-~---_ ..-
:c
r, !" ...':,j
IOOr'---~-
,2.:
·;;;'CC)
;;J
c51..
~1~_.
...7:<_... :,r
~?
LJ(.i<~l
Figure 11.10 Solid Lines: Proposed Mean Damage Ratios for High-Rise BuildingsDesigned for Different UBC Zones (after Whitman, 1973);
-Dashed Lines: Mean Damage Ratios for High-Rise Buildings in Los Angeles(after Whitman and Hong, 1973);
-Dotted lines: Bounds to the Mean Damage Ratios for Light Industrial Constructions(after Benjamin, 1974)
39
INTENSITY r (MM)
X
*~.IO
IX9
~J
5CYYc:J-----4------+-----71c...;.;::
VI VII
6 7~*3
'OO'YcE==t=-==_=---.-~=_=--=~=~--~-=_~_~~_==-$=i==;;~~~E~
.....o:::E
oI«0::
@ Son Fernond0
o Anchorage
-+ Kerns
o San to Bar bora
X Santo Rosa
\l Caracas
U1L*' San Francisco
REFERENCE QUAKES1'(- I No reports have been found of major skeleton frame
collapse during th" :906 SanFranclsco or 1923 Toykoq\.ulKes.
1<" 2 De,;igned for Zone 3.
@@AVi'roge damage ratios
8-0 Class 8 - wi th no earthquake loading.
A-3 Class A -- with Zone 3 earthquake loodino_
*:3 Collapses shown above 100% line.
* 4 Reference 25
Figure 11.11 Mean Damage Ratios for Skeleton Frame Structures(after Mann, 1974)
* C]ass A - Steel frame structures
* Class B - Reinforced concrete frame structures
40
III. ANALYSIS OF UNCERTAINTY: TYPES
The frequency-magnitude and the frequency-intensity laws presented in
Section II (Equations 11.1 to 11.8 and equation 11.13) are idealized relation
ships, fitted to historical data. The same is true for the intensity-expected
damage curves in Figures 11.5, 11.6, 11.8-11. The quantatitive limitation of
statistical information is not the only problem of seismic inference. Additional
difficulties are due to often present biases and to the incompleteness of
historical records. Uncertainties on MMI data include: the uncertainty on
epicentral intensity, which may be higher than the intensity at the closest
inhabited center; the uncertainty on the mean rate of events in the low-to
moderate intensity range: the older the record, the less complete the data;
the uncertain effect of neglected local soil conditions; the uncertain effect of
aftershocks, which are typically removed from the statistics; the uncertainty
on the epicenter location and on the focal depth.
There is also reason to believe that damage statistics collected through
questionnaires are inaccurate and biased. On the other hand, di.rect subjective
evaluations of damage, such as those in Figures 11.8-10, differ from author to
author.
Because of all these sources of uncertainty, only limited confidence can
be placed on anyone probabilistic model which is estimated from statistical data,
or which relies on professional judgement. In some cases (e.g., in the
estimation of the seismicity parameter b for California) the data base is so
large that statistical uncertainty can be neglected in the context of the overall
accuracy of the analysis. In other cases (e.g., in the estimation of the
seismic parameters in a low-seismicity region, or in establishing the resistance
distribution of a new piece of equipment) statistical variability may be a major
source of uncertainty and risk.
In Sections IV and V, seismic risk models will be classified into two
categories: (i) models which result from best data fitting (or from other
statistical estimation procedures), and which do not include inductive uncertainty.
These models are called "probabilistic," and will be studied in Section IV;
(ii) models which incorporate inductive uncertainty; these models are called
"statistical," and will be studied in Section V.
Although probabilistic models can be vieWed as limit cases of their
statistical counterparts,as the amount of information "tends to infinity," they
41
are considered separately on account of their greater simplicity. Also, most of
the seismic models proposed in the past have been of the "probabilistic" type
(for exceptions see Benjamin, 1968; and Esteva, 1969). It is found appropriate,
therefore, to quantify the effects of statistical uncertainty through penalty
factors on the "probabilistic" mean failure rate.
The theory of statistical prediction (of future random events, under limited
information on the generating probabilistic mechanism) has been developed mainly
in the last decade (Thatcher, 1964; Aitchison and Sculthorpe, 1965; Guttman,1970).
Different methods and different terminologies are used, depending on the meaning
of probability, and on the inference school (frequentist, likelihood, fiducial,
Bayesian). Preference is given here to the Bayesian viewpoint, but the numerical
results can be readily given a frequentist, or a likelihood, or a fiducial inter
pretation. The general methodology and some specific results to be used in
Section V are reviewed next. For a more detailed account of the theory and for
applications in the area of reliability, see Veneziano (1974, 1975).
Consider a random vector X (for the case of interest here, ! might include
some measures of site intensity for the next earthquake and some resistance
parameters of the facility at risk), with distribution function FX(·). Suppose
that the type of distribution in known (this assumption can be released, see
Veneziano, 1974), but that uncertainty exists on some of the parameters (for
example, on the mean value vector, on the covariance matrix, etc.) If e is
the vector of unknown parameters, with Bayesian distribution Fe (·) ,and F!le(·)
is the conditional CDF of !, the unconditional distribution of X is, from
the total probability theorem:
(IlLl)
In general F!(.) and Fxl£(·) differ both in the parameters, and in the
distribution type. In fact, it is precisely this condition which differentiates
probabilistic from Bayesian-statistical models.
The probability distribution Fe (·) in Equation (1) can be either the "prior"
distribution Fe(·), or the "posterior" distribution, Fe(o), the latter including
information in-addition to that already contained in Fe. If z denotes this
additional information, Fe can be found from Bayes' theorem:
42
where 1(8Iz)OCfz!Q(z/Q) is the likelihood function of the experiment which
generates z.
For applications in Section V, consider the special case of a normal random22variable X-N(~,a ), with unknown mean ~ and/or unknown variance a. Information
on the unknown parameter(s) is provided by a prior distribution and by the random
sample ~={Xl' X2, ••• ,Xu} from the unknown population of X. The problem of finding
the predictive distribution of X, Equation (1), was discussed, among others, by
Raiffa and Schlaifer (1961) and by Guttman(1970). The results given below are
for the case of the unknown parameter(s) having conjugate prior distribution
(Raiffa and Sch1aifer, 1961). Under this condition the Bayesian results are
numerically identical with the frequentist results obtained by Proshan (1953),
after an appropriate redefinition of the sufficient sample statistics.
(a) ~ unknown, 2a known
For a normal prior distribution of ~: ~-N(~';a,2=a2/n'), the posterior
distribution of ~ is also normal:
/" '" N (ji" ;
where II
/ ::::., .
)" In = n+1:1
From Equation (1), X has normal posterior predictive distribution:
(b) ~ known, 2a unknown
(III.2)
From Raiffa and Sch1aifer (1961) the family of conjugate distributions of
the precision parameter h=1/a2 is Gamma-2: for a prior density in the Gamma-2
form:
, h>o,2-
) 11' ) S > 0 J
the posterior density of h is:
43
where2
II
S =2 2.
n' 5' + 11 f1-----_._._--J1' + n
and 2. 1. ",1'1 !2.
S = n L (Xi -;U)i=1
In this case, Equation (1) yields a predictive tn,,-distribution for y=(X-].1)js",
with density:
(c) Hand oZ unknown
~ (~~)
r (11'/2.)( 1 ,)- Cn \1)/2
1. + Lj n '.. . (III. 3)
From Raiffa and Sch1aifer (1961), the conjugate family is now Normal-Gamma.
If the prior parameters are [n l, ].11, (n l -l) SIZ], this means that
Given the sample {X1 , ••• ,Xn }, it is found that the posterior distribution of
(].1,02) is also Normal-Gamma, with parameters [nil, ].1", (n"-1)S"Zl,where
nil = n'+u
n5
2= _1._ Z eX- _P)2
1:1-1. - i 1J.-=
Scu1thorpe (1965) found that
j-L" = (11ju' + nrJ!n"u I? I ,2. 2-(n -1) S = (n - 1.) S + (n -:1) S -+-
and 0, sZ are the sample statistics:n
A 1 '\? =: - L Xin . :11=
From Equation (1) Aitchison and
Inn:n'1
n'l )1/2. X _~ II
11" -+- 1. 5
has t " I-distribution; i.e. that the prediction density of y has the form (111.3),n -
with (nil-I) replacing n".
In Section V it will be shown that replacing the suggested normal distribution
of the seismic resistance expressed in terms of In a or of MMI (see, e.g.,
Newmark,1974,and Benjamin,1974) by a (predictive) t-distribution may increase
considerably the calculated risk.
44
IV. Probabilistic Seismic Damage Models
The information on seismic risk and on seismic resistance of engineering
systems reviewed in Section II is used here for mean failure rate calculations.
A simple, yet realistic model is presented first, for which closed-form results
are readily obtained. Thereafter, more sophisticated models are introduced and
studied numerically. In all cases inductive uncertainty is neglected. Statist
ical (inductive) versions of the same models will be considered in Section V.
IV.l Linear Gaussian Model
Consider the "linear" damage model in Figure 1 (lower part). D denotes
the actual damage or the actual damage ratio, and df is the value of D at
"failure." For each given MM intensity I, the probability distribution of
Log D is assumed Normal (as suggested by Benjamin, 1974), with mean value
aD+bDI (see Equation 11.16) and variance a~. Then the probability of failure
for an earthquake of site intensity I is:
(IV.l)
where ~f'] is the standard normal CDF. Pf(I) is also the probability that the
resistance (with respect to the threshold damage df ) is less than I, meaning
that the probability distribution of the resistance R (in units of MMI) is normal:
R"'-- N (JAR ; CT~)
with mean value: rR = (d.£ - aD)/bD ~
and standard deviation: CTR =: CTD/b D~D- i
(IV.2)
(*)(see Equation 11.15)
Typical values of SD are given in Table 11.5. In terms of the normalized
intensity IN' defined:
(*)The use of Equation (2) in the following calculations is numerically
correct, but the reader may disagree on its interpretation as a resistance
distribution.
45
(IN measures the algebraic distance of I from the mean resistance in units of
° ), R is a'standard nor~al variate, R-N(O,l). Let A. be the mean rate of eventsR 1
with site intensity larger than i. In its simplest form, the model assumes that
A. varies exponentially with i (See Figure 1, upper part):1
A-l
e (IV.3a)
For typical values of SI see Equation (11.13) and related comments, Figures 11.1,
11.2, and Table 11.3. Alternatively, in terms of the normalized intensity
iN=(i-)lR)/oR,Equation (3a) can be written:
(IV.3b),-SI)lR
where A =Ae is the mean rate of events with site intensity larger than theo
mean resistance, and
~N :=. ~I CTR = f3T /(3D
For SI=SIo=slope of the frequency-epicentral intensity relation (see Equation 11.12
and lines between dots in Figures 11.1 and 11.2), and using Tables 11.3 and 11.5,
SN is found to vary between 0.60 and 1.20, with typical value of about 0.90.
The mean failure rate, Af , can be calculated from Equation (1.3), which in
(IV.4)
the present case becomes:
Af~ Ao f 00 e -(3N il'J
viTI - d:)
S2/2The quantity yDET=e N can be interpreted as a penalty factor for uncertain
resistance (Le., with respect to the "deterministic" case 0R=OD=O); it increases
with OR and with SI (see Equation 3b), and is 1 whenever either of these
parameters is zero. Typical values of YDET are in the range 1.20 to 2.05.
However, it will be shown later in this section that YDET may increase when one
allows for upper truncation or for other nonlinearities in the frequency-
epicentral intensity law.
The exact mean failure rate, Af , can be compared with sometimes used
approximations of the form (1.4), rewritten here:
(IV.5)
46
where FR P is the P-fractile of the resistance distribution (here. of the•
standard normal distribution; i.e .• FR.p=~'P)' Recall that Afp is the product
between AoexP(-SNFR.P)' which is the mean rate of events with intensity
I>ip=~.paR+~R' and P, which is the probability of failure if an earthquake of
intensity i p occurs. Values of P between 10-1 and 10-2 (USAEC Reactor Safety
Study. WASH-1400. Preliminary Report) and between 10-2 and 10-4 (Newmark. 1974)
have been used. In Figure 2. the ratio
1
P (IV.6)
is plotted versus SN for selected values of P. For SN=l it is Y10-l=4.6;
1Y10-2=16.l; Y10-3=75; Y10-4=400. The curve YDET- 2 YO•5 gives the factor of
unconservatism when the resistance is assumed deterministic and equal to its
mean value ~R'
IV.2 Nonlinear Gaussian Models
Five cases are considered: (a) truncated linear frequency-intensity law;
(b.c) truncated linear first and second derivatives of the frequency-intensity
law; (d) quadratic frequency-intensity law; (e) logarithmic frequency-intensity
law.
(a) Truncated-Linear AiN
The resistance of the system is modeled as in the previous case (Figure 1.
lower part). but now the frequency-site intensity curve is truncated at the
upper bound intensity level i 1 (See Figure 3. curve b). As observed in
Section II. this is a good approximation to calculated site intensity risk curves
when the epicentra1 intensity is bounded (See Figure 11.1 and related comments).
For mathematical convenience. let i 1 be the algebraic distance of the
upper bound from the mean resistance. in units of standard deviations of
resistance. In this case. i1~ for untruncated site intensity. i1=0 for
truncation at the mean value of resistance. Formally. the mean rate of events
with (normalized) site intensity in excess of ~ is:
~{:0 -rn iN for iN :{. 11.e ,/\. (IV. 7)
IN for ., IN > 1.1 ,
47
which replaces the risk law (3b). Using Equation (7), the mean failure rate is:
A£ .1\0 t: -n.iN - i;/2
oLiN- e,li 12 TTl
2
AofN/2
~ (i 1 + (3N) (IV.8):=. e .
For Af defined as in Equation (5) (F =~ =P-fractile of the standard normalp R,P 'pdistribution), the ratio
iP
(IV.9)
is plotted in Figure 4 as a function of iI' for SN=l.O, and for selected values
of P. It is emphasized that Afp is calculated as if the risk curve were not
truncated at iI' which fact makes Y . defined also foril<~'p,P,1.l
As il~' the ratio (9) approaches Yp in Equation (6) and Figure 2. Indeed,
values of Af,il very close to Af are found for il>O, meaning that truncation of
the risk curve above ~R has little effect on the calculated mean failure rate.
This in an interesting conclusion, which shows that Af is not always sensitive
to the decay of the seismic risk curve in its upper "tail," as commonly believed.
From Equation (8) it is apparent that the upper truncation point for which the
mean failure rate becomes half the value for no truncation is: il=-SN (il=-l in
Figure 4). The penalty factor YDET,i ~O.5,il applies when Af is approximated
by Ao (i.e., when assuming GR=O and wken using the untruncated linear model,
Equation 3b).
Notice that all the curves in Figure 4 are obtained by simple vertical
translation of the curve YDET,il' The same being true for any fixed SN' it
is convenient to plot the factors YDET,il
for several values of SN (Figure 5),
and to tabulate separately the factors by which YDET,il
must be multiplied to
calculate Yp ., (This is done in Table 1 for selected values of P.),1.1
Example. Five linear approximations to the risk curves in Figure 11.1 are
shown in Figure 6. Some of them are untruncated, and correspond to sources with
no (or very high) upper truncation of the epicentral intensity. The other curves
are truncated, being approximations to those frequency-site intensity relationships
in Figure 11.1 which used small or moderate upper bounds on the epicentral
intensity. The values of SN' Ao and i l for the five approximations are given
in columns 2,3 and 4 of Table 2, respectively. For a normal resistance
on the frequency-epicentral intensity law have
mean failure rate through variations of A ando
on these seismicity parameters will be considered
48
distribution with mean ~R~8 and standard deviation 0 r =O.8 (see Figure 6), the
exact mean failure rates, from Equation (8), are given in column 5 of Table 2.
The same values could be found from: Af,il=Ao·YDET,il' the last factor being
plotted in Figure 5. Finally, the last two columns of Table 2 refer to the
approximation (5), where FR,p=¢'p and P=O.I, 0.01, respectively. The numbers in
parenthesis are the factors of unconservatism, Yp,il' associated with the
approximations (see Equation 9, or Figures 2,4,5 and Table 1). It is observed
that:
(i) truncation of the frequency-site intensity law has a small effect
on the mean failure rate for il>O. However, the effect would
increase markedly for truncation values il<-SN; the latter is the
case for very reliable systems (for high ~R).
(ii) truncation of the frequency-epicentral intensity law is more
important, primarily because it reduces the mean rate A. (Ato
the same time it increases BN
and causes a sudden drop of the
risk curve at the site);
(iii) the factors of unconservatism associated with the approximation (5)
are not sensitive to truncation of either the epicentral, or the
site intensity laws.
Since different assumptions
sizable consequences on the
BN, statistical uncertainty
in Section V.
It has been observed (Cornell, 1975) that the truncated linear model (7)
is logically unsatisfactory because it associates a finite mean rate (namely,
Aoe-SNil) to events with site intensity equal to the upper bound i l • (This does
not mean, however, that the model should be avoided as a mathematical approxi
mation.) Several other models with upper truncation can be formulated, which do
not display this singularity; four of them are considered next.
(b) Truncated Linear dAiN/diN
By truncating the first derivative of the linear risk function (7) at iI'
and by imposing the condition Ail=O one obtains the following frequency-site
intensity law:
49
(IV.lO)
(IV.H)
(see a representative plot in Figure 3, curve c). When used in Equation (1.3),
the risk function (10) yields the following mean failure rate (compare with
Equations 4 and 8):
A£ . - AO[e~:/2. 9? (it + ~N)- e~~i.i~ (iL)J.'~1.
One might still argue that the model (10) implies a discontinuity in the
mean rate "density" at i l (from the value AOSNe-SNil to zero), and therefore
that it is also physically unsound. A "better" model might be obtained by
truncating higher order derivatives of the risk function. Truncation of the
second derivative at i l generates the following model (for a representative
plot, see Figure 3, curve d):
ll.N= {/..00 [( e-f3NiN
- e-~Ni1)_ ft(i1 _iN)e- f3Nii], iN < ii
1\ - (IV.12), lN~i1.
which gives the mean failure rate:
where ¢(.) is the standard normal density function.
In both models (10) and (12), Ao is the mean rate of events with site
intensity in excess of the mean resistance, for the case of no truncation,
il~ (see Figure 3).
A comparison of the three "linear" models with truncation, Equations (7),
(10) and (12) is straightfoward in terms of the mean failure rates, Equations
(8), (11) and (13). However, when using different truncated "linear" models to
approximate the actual nonlinear frequency-intensity law, one would conceivably
select different values for iI' and possibly for Ao • In so doing, one would
reduce the difference between the risks calculated from the various models. In
the remainder of this study, no further consideration will be given to models of
the type (10) and (12).
50
(d) Quadratic
A quadratic law for Richter magnitude was proposed by Shlien and ToksBz
(1970), and by Merz and Cornell (1973). A quadratic model is used here to
approximate frequency-site intensity curves (such as those in Figure 11.1 and
11.2). Let then:
(IV.14)
where, as in the linear case, A is the mean rate of events with site intensityo
exceeding the mean resistance value, and aN,SN are known parameters. Equation
(14) is an appropriate risk function only if it is non-increasing; i.e., only
for iN>-SN/2~.
For the case of no upper bound site intensity, the mean failure rate can be
calculated analytically:
(IV.15)
Integration over the entire real axis violates the condition that AiN should be a
non-increasing function of iN. However, if failure events caused by earthquakes
with normalized site intensity iN<-SN/2aN are negligible, the mean failure rate
(15) is numerically accurate. When ~=O, Equation (15) reproduces the mean fail
ure rate for the linear law, Equation (4).
It is interesting to compare the exact mean failure rate, Equation (15),
(IV.16),
with a conservative approximation obtained from a tangent linearization of the
quadratic law. Linearization around iN=~ yields:
- J3N i"" . iNAo ."* . e • N.. IN
where
., ,and the following upper bound to the mean failure rate:
(IV.17)
(IV.18)
in Equation (16), which means that
which produces the least upper bound is found for
51
The tangent approximation* .iN=-SN/(2~+1), being:
A .I'c f=111in (Af .*) = mUl 0, J.
N ex p (-- 0v i *. iN - i: /2) cli wi': ,1N i: l{iTf -Db ' IV
A0 e x p [(3N.2/ ( 4 ~N -+- 2)J .* * .*This choice of iN corresponds to S .=-1N,:'-N N
the maximum contributions to the risk for the quadratic and the linear tangent
*laws occur both at iN=~ , and that such maximum contributions coincide (evaluate
the integrands in Equations 15 and 18 for iN=i~).
The ratio
is the factor of conservatism for the "best" tangent approximation. Risk curves
with BN=1.6 and ~=O.2,O.3 are shown in Figure 7. The factor of conservatism is
1.18 for eX N=O. 2, and 1. 26 for dN=O. 3, showing that in this (realistic) range of
~N values the tangent approximation produces accurate results. These calculations
also suggest that accurate linear approximations to nonlinear seismic risk curves
*can be obtained in general by choosing the point of tangency, iN' so that the
* *derivative at iN' -BN,i;' equals ~.
(e) Logarithmic
Consider the 3-parameters frequency-site
cL [In Ci i - iN) - In cJ/\i = A e
N
intensity relationship:
c,d,)., >0(IV.19)
where i l is the intensity upper bound, c is the value of (iI-iN) for which
AiN=A, and A and d are a location and a scale parameter on semilog paper,
respectively. (Note that A is a redundant parameter, which is introduced only
for mathematical convenience.)
1. // 01 - f(. .) (i)11.- 1 N
The "logarithmic" law (19) (plotted in Figure 8
corresponds to the Pareto distribution of (il-iN)-l:
.Jec l
for c=l and d=1,2,3)
Reasons for using risk functions in the form (19) are:
52
(i) they include an upper bound intensity;
(ii) the mean rate of events with site intensity in excess of iN is a
decreasing function of iN"
Neither of these properties is enjoyed by the quadratic law (14).
For a resistance distribution R-N(O;l), the mean failure rate
(IV.20)
(IV.21)
is a function of iI' c and d. The linear approximation with slope numerically
*equal to the intensity ~ at the point of tangency is found for
i: :1.1 - ,; it + 4 cL ' (= _ ~ ."*) ,2 N,lN
with associated mean failure rate:
Af , i N7(- A[(i 1- - i ~ )/ cJd- ex p[- d i:I (i i - i:) + ft: i ~ /2]-LC . r;-------:-l)I JcL (3 .* 2.)/I 1;1. + --V iJ. + 4 cL Z C e)( p 2 IN •
(IV.22)
The factor Af,i~/Af' by which Equation (12) is a conservative approximation to (20),
depends only on d and il; it is plotted in Figure 9 for d=1(1)5 and for i lvalues in the range (-3,3). For truncation intensities which are not much smaller
than the mean resistance, the linear approximation (22) is quite accurate.
Clearly, in the actual linearization of convex risk curves one should not use a
tangent approximation, if not to calculate upper bounds for Af • Figure 9 shows,
however, that for logarithmic risk functions the tangent upper bound is itself
quite close to the exact mean failure rate.
IV.3 Linear Gamma Model
Suppose now that the normalized resistance R (zero mean, unit variance) has
shifted Gamma distribution, with density:
-D(r+D)e
, r < - D
,y?--D,(IV.23)
D=positive constant
Plots of the density (23) are shown in Figure 10. Being 0R=l, the Gamma density
(23) reduces to a shifted exponential when D=l, and to the standard normal N(O;l)
as D-7<X>.
53
For the untruncated linear law (3b), one finds a mean rate of failure
~f =
(IV.24)
The ratio between the mean failure rate for Gamma (Eq; 24) and for normal (Eq.4)
resistance distribution:
(IV.25)
is plotted in Figure 11 as a function of SN' for D=1,2,3,5,8. The ratio (25) is
generally smaller than 1, due to the Gamma density (23) vanishing for R<-D.
This shows the importance of the left tail of the resistance di.stribution, a fact
which will be fully emphasized in the following section. However, with the
exception of the rather artificious cases when D<l, and for typical values of
SN (say, SN<1.5) the mean failure rate does not change significantly if one
replaces the normal resistance model by a Gamma model with the same first two
moments.
FRACTILE SNp 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00
0.5 2 2 2 2 2 2 2 2
10-1 4.63 3.59 2.77 2.15 1.66 1.29 0.995 0.770
10-224.77 15.55 9.77 6.13 3.85 2.42 1.52 0.954
10-3 156.6 84.42 45.50 24.53 13.22 7.13 3.84 2.07
10-4 1074 510.4 242.6 115.3 54.80 26.05 12.38 5.88
10-57738 3298 1405 598.8 255.2 108.7 46.34 19.75
Table IV.1 Factors by which the values YDET,il in Figu~e IV.4
must be multiplied to obtain the ratio yp' in Equation (IV.9),11
Col. 1 2 3 4
55
5 6 7
RISK SN=SIoOR;\(*) i 1
;\ (*) ;\ - (y (~» ;\ (y (~»CURVE
0 f,i1 f,P-0.1 0.1,11 f,P=O.Ol 0.01,11
1 0.88 5.9 - 4 co 8.7 ~ 4 1.8- 4 (4.7) 4.5 - 5 (19.1)
2 1.55 8.6 - 5 co 2.9 _. 4 6.3 - 5 (4.6) 3.2 - 5 (9.0)
3 1.55 8.6 - 5 0 2.7 _. 4 6.1 - 5 (4.4) 3.1 - 5 (8.6)
4 1.58 2.6 - 5 00 9.1 _. 5 2.0 - 5 (4.6) 1.0-5 (8.8)
5 1.58 2.6 - 5 -0.625 5.3 _. 5 1.4 - 5 (3.9) 7.3 - 6 (7.3)
(*) -nNotation: X-n=X·10
Table IV.2 Mean Failure Rates for the Risk Curves
and the Resistance Distribution in Figure IV.6
T
--I
56
iMODIFIED MERCALLI INTENSITY, I
57
10
DET
increasing,6N=Br 'OR
~------,-II
P I
]JR I
2.01.81.61.41.21.00.8 6NFigure IV.2 Untruncated Linear Frequency-Intensity Law
0.61
Ratio Yp=Af/Af in Equation (IV.6)p
58
SAo
Cd)
in-I;... (1
(a)
-3 -2 -1 a 1
Figure IV.3 Untruncated and Truncated "Linear" Risk Models;
See Section IV.2
,)
l.OL.
1
,_ -.L,.V
-2.5 -2.0
59
-1. 0
NOR11ALIZED UPPER BOUND
SITE INTENSITY, i,1.
o 1.0
Figure IV.4 Truncated Linear Risk Cu~~e~ Ratio
(Equation IV.S) 6N=1.O
SN=2.00
----- 1.80~
~ ------ 1.60~
~r- / - 1.40
V ~V~~
-----~ 1. ;;0
V ~~",,-~ 1. 00
~~~ C.80.
/ ~ QJiQ._.-./ " /' A" --- --- SN=O/ / ./ ~ --ioo""'"
----"'"" ~!/ ~~ ./ ./' ~ ~ ----/ ./ /' " ./..... ",
/ /' /' / /' ",-
V / /' /' ,,' --/ ~
r- / //// /' -~
V/V/~/r
/ -~/
VIV ~'
/ -
/~
f ,f f /
/ / ~, / I
/ ~'/ ~', .
r- I -
10
5
0,1
0.03-2.0 -1. 5
60
-1. 0 -0.5 o 0.5 1.0
?,OhMALIZED UPFER BOFND SITE INTENSITY, .1.1
Figure 1\7.5 - Truncated Linear Risk Law. Ratios YDE~ . =, 1. 1-
132/2 ' 1\f . n, =e N ¢(i1+I3
N) for different values of SN'
,11
0
61
10-2 ~~....--+-------+-----+----+-----l
10-3r: q 1()-4....... X..L.J.L _
.-4 r:10 ~ 06 ,1:.[H'"'-------+-~l.__-~+_---_l_--.300.~__l
2.6xlO-
10- 5 1------+-----+----I---I-~k__-__l_~--__1
v VI VII .f.l =VIIIR
IX x
HHI at Boston (Firm Ground)
Iili.l:'rE~ IY~-l,inear ilE.c:L.Truncated.-l-inear ~!,oxima_tions to _t:I1l:'Risk Curves in Figure 11.1
62
10
1
i\ i lieNt 0
-110 - I-------t-----+--------j~---_+_-----~~~-_t--.:._~,..__~
10-2I------+----+-------il------+-------l--------I~r___-~.....J
J.0- 31-- ..L- ...L.- -L --l.. --L ---l.__---lL.....J
-4 -3 -2 -1 a 1 2 3
NOID1ALIZED SITE INTENSITY, iN
Figur_e IV.? Quadratic Law (IV.l!~);
UN=O.2, 0.3; BN=1.60. The dashed-dotted lines are linearizations
around i:=-B,,! (2aN+l); see Equation (IV.18)
J.'~ _--::..:-.- _
310
.1
-7 -6 -s -4
63
-3 -2 -1 o
Figure IV.8 Logarithmic Frequency-IntensiiY~~~IV.19);
G=l
64
d=2
d=3
d=4d=5
d=l
3
1
1 /.J.:1. ''-7
-3 -2 -1 o 1
Fi&ure IV. 9 ~ogarithmic:- Law....!.. Conserva tism of the.
Jange~t Approximation (IV.22)
0'VI
3,>.c.1o
I
D=1 I
(exponential)
-1- 2-2-3
1
o
D=-ff
0.5 I- D-~~\IZ
D: ; " ~1J:.t:l
~
Figure IV.IO Shifted qamrp.a~~nsit:ies, Eq.(Ii{.23).
In all cases the mean is zero and the variance is.1..
66
GA}~ ~~NORMAJ~
0.2 D
1.81.20.6
0.6
0.8
1.0
bN
Figure IV.II Ratio between the Mean Failure Rate forGamma and for Normal Resistance Distributi~ns,Eq.(IV.25)
6 ',7! ,<
v. Statistical Seismic Damage Models
The models analyzed in the last section are intended to be "best" estimates
from statistical data. Unfortunately, the information available on the seismic
risk at a site, and even more on the resistance distribution, is far from
supporting conclusively any particular modeL As a result, both the "correct"
type of the model (e.g., whether the risk law is linear, or logarithmic, or
other; whether the resistance distribution is normal, or Gamma, or other) and
the "correct" parameters values (e.g., the mean occurrence rate A and theo
slope SN of the linear model) remain uncertain. In the same sense, the parameters
of the normal or Gamma resistance distributions are essentially unknown. A few
models, in which inductive uncertainties on the parameters are taken into consider
ation are studied in this section. Some of the numerical results are reported
in Appendices A and B.
V.l Uncertainty on Demand Parameters
In Section IV it was shown that the linear law:
(V.l)
or a truncated version of it provide accurate approximations to calculated
nonlinear risk curves at a site. The parameters 1..0
and SN depend on the
regional seismicity, on the assumed upper bound epicentral intensity, and on
the attenuation law. Following the general Bayesian approach in Section III,
1..0
and/or SN are considered now to be random variables, with given probability
distribution. In each of the cases studied, the effect of inductive uncertainty
is quantified through multiplicative penalty factors on the mean failure rate
under perfect statistical information.
(a) Linear Gaussian Model; 1..0
unknown; SN known
Let N(O;l) be the probability distribution of the (standardized) resistance
R. If A in Equation (1) has lognormal distribution:o
(V.2)
68
and SN is known, the mean failure rate is
(V.3)
which means that statistical uncertainty on Ao increases Af
by the (penalty)
(V.4)
is the ratio°(Notice that e lnAoin Figure 1.
A£If3N
Af 1/\0' f3NcHnAversus e 0YA is plotted
obetween the values of Ao at (E[~nAo]+OlnAo) and at E[lnAo])' In the range
eOlnAo=0.4 1 1 ( h' d 'to • w ~ch correspon s to I-sigma uncertainty factors on A ofo
factor:
1.5 to 3), the penalty YAo varies from 1.1 to 1.8.
(b) Linear Gaussian Model; Ao known; BN unknown
2Now let Ao be known, and BN have normal distribution N(~SN;OBN)'
R~N(O;l) the mean failure rate is found to be:00
).Wo = LAr IAo , ~ • J F~N (r,.,)00
~<Tr,~f ex p [~ [i;,"(f,,-r~JI<Tr:J Jttv -00
2(In all practical situations,it is o~<1.)
For
The associated penalty factor with respect to the case 0BN=O
2. 2 ( .2
(
2 )-11-2 1~a UP. / 1 - cr t1,)~ 1.- u0. e N
IV
is:
(V.6)
In Figure 2 this factor is plotted versus ~B for OQ =0.1(0.1)0.5.N I-'N
Typical values of ~SN are between 0.8 and 1.6, and of USN between 0.1 and
0.3. This implies a typical YBN range of 1.01 to 1.2 (the upper limit of this
range is, however, very sensitive to the assumed maximum for USN)'
(c) Linear Gaussian Model; Ao and BN unknown
In general, both Ao and Bn
are unknown 0 For the marginal distributions
69
given above and under the condition of independence the mean failure rate is
with
.::1.
l!1- CTj3~
associated penalty factor:
(V.7)
y . ¥~/\ 0 I~N (V.B)
A convenient visualization of what this distribution
f«ln:~NO-fN])- (V.9)
implies (and a convenient
YAo and¥SN as in Equations (4) and (6).
In most practical cases the assumption of independence between Ao and SN is not
appropriate. Let then In Ao and SN have generic bivariate normal distribution
. [u{nAo .) f u1 n AO ut3i\.-
means of selecting the parameters of the covariance matrix) is suggested in
Figure 3. In the figure, a (normalized) intensity level i d is defined, such-S i
that the mean rate Aid=AOe N d is independent of SN' For example, in the case
of Figure 1101, the condition of independence might be satisfied at MMI 4 or 5,
which implies a value (4-llR)/OR or (5-:llR)/OR for i d (llR and OR are parameters of
the resistance distribution).2If the mean and the variance of In Aid are denoted simply lld and ad' the
joint distribution of In Aid and SN is:
L~Ai~ ~ N(~~ J.[:J :J). (V.IO)
with
i:,o-~]).(3N
In general it is id<O and In Ao and SN are negatively correlated.
From the joint distribution (9), the conditional distribution of (In AoISN)
is ea$ily found:~
(In Ao I~N) ~ N (l'lnA o + f ;r:;>'o (PN -/'fd ; (1.- p2) a-{n?J .
Then, using Equation (3):
(V.12)
70
the unconditional mean failureintegration with respect to PN yields. 00
Af = j '\fl~' f~ (r~J J ~N_ 00 N N
i
Finally,
rate:
~ ~ - cr 0,.", I
which can be written, in the notation of Equation (11):
ex p {ttd. + icLjA~N + i ((JJ: + icf cr~N)
+ %()APN + iJrJ;Nfl (:i-(j~N))· (V.14)
From this equation, and after some algebra, one can express the penalty factor
on Af due to statistical uncertainty of Ao and SN:
,(V.lSa)
where: YA.J.d
is given by Equation (4) with Aid in place of Ao ; see also plots
in Figure 1;
YSN is given by Equation (6) and is plotted in Figure 2;
(V.lSb)
For id=O, Ao and SN are independent, so that Aid=AO and Equation (lSa) reproduces
the results (7) and (8). If the mean rate Aid is known with certainty (which
means that the uncertainty on Ao is totally explained by ~) the penalty factor
(lSa) reduces to: YA p, = 1"13 .YioL . If in addition it is id=O, Ao becomes0, IN J N
known and Y). (.l = YQ .0,1'4-- ':-v
The factor Yid depends on i d , ~SN and crSN• Plots of Yid versus ~SN for
i d=-1(-1)-8 and crSN=O.l(O.l)O.S are shown in Figure 4. For small lidl, Yid
is not sensitive to ~SN and crSN' and is generally smaller than 1. As lidl
increases Yid also increases, with high penalties for combinations: crSN large,
~SN small. For ~SN in the range 0.8 to 1.6; crSN in the range 0.1 to 0.3, and
for id=-S, Yid has values between 1.05 and 2.3. (More will be said on the
selection of the seismicity parameters in Section VI.)
Cd) Linear Gamma Models
Suppose now that the nOllualized resistance (zero mean, unit variance) has
71
the shifted Gamma density (IVo23) and that the seismic risk at the site has
the linear form (1), with one or both parameters unknown.
* SN known, and ln Ao~N(~ln Ao; 02ln Ao). Integration of the conditional
mean failure rate with respect to Ao yields:
~£I~N::: Jo;flAoJI?v . fAo (AJ cL;to2.
=0 ( D : (3A/ ) D ex: p Ifi D +lIn Ao + 1z O--l~ AD 1 . (V.16)
By comparison with the mean failure rate for Ao and SN known, Equation (IV.24),
the penalty factor YAo is found to be the same as .for normal resistance, i.e.,
Equation (4).
* If Ao is known, and ~N has Gamma distribution G(K,r) with density:
K-:1. r(3r (r (3,v) e - Iv
rCK), (V.l7)
the use of Equations (IV.24) gives the following expression for the
where the symbol
mean failure rate:I<
).. _;\0 rflA o - r(K)
= ?Nf<-~
fa (2+ ~t( ~2.; 1. ; n) (K; 1 ; n)
111 [D(r-D)Jl1.(V.18)
(m;d;v)=m(m+d) (m+2d) •.. (m+(v-l) d) v=1,2, .••
For a generic density function fSN(SN), the same mean failure rate must
be calculated numerically from
However, for practical purposes, it is not 'very important which distribution
one assumes for SN' since Af/Ao is typcially close to the conditional mean
rate AfIAo,SN=~SN (see Equation IV.24). This qualitative conclusion is in
agreement with earlier results for normal resistance and normal distribution
of SN ; see, e.g., Figure 2.
72
* Assume now that AO and SN are correlated random variables, with
. distribution: BN-C(K,r) and (In AOISN)-N(Vo+id(SN-VSN); a~). As
already discussed for the normal model, this corresponds to In Aid=
(In AO-SNid) being independent of SN' with distribution:
N(Vo-idVSN;aa)·
Conditional on given SN the mean failure rate is:
])2
A£I~ = (J)~(3N) expVo .-icLj'l.~+ ±erl+ (icL-t-D) f?v}. (V.19)
Integration with respect to SN yields the unconditional mean failure rate:
At == ICOAt jf1 . f rN erN) cL ~4/
If AO and SN are independent, then
the penalty factor for statistical
Equation (18).
i crJid=O and Af='YAo:AfIAo, where YAO= e2.. is:
uncertainty on /\0' and AflAo is given by
V.2 Uncertainty on the Resistance Parameters
One way of introducing uncertainty on the resistance parameters is to
treat aD and the constants aD and bD in Equation (11.16) (see also Figure IV.l)
as Bayesian random variables. In this study, the simpler approach is followed,
of quantifying statistical uncertainty directly on the parameters VR and ai of
R. Under the assumption that the actual distribution of R is normal, use will
be made of the statistical prediction results in Section III.
(a) VR unknown, a2knownR
A 2Let VR-N(VR,aR/n). In a Bayesian approach this distribution corresponds,
for example, to a random sample of size n being available from the population
of R, and to a noninformative prior distribution of VR (to n'=o in the results
of Section III). The same distribution would result from the model in Figure
IV.l, if aD and 1D were known, and ~ were estimated from n independent data
points.
Under these conditions the predictive distribution of R is, from Equation
(111.2), N(~R;(l+l/n)a~). and the reduced variable Rj=(R-CR)laR~l+l/n'has
73
is the mean rate of events with sitestandard normal distributiono If AoA
intensity larger than VR, the mean failure rate is:
{3",,2 (1-r i (1'1) /2AI lOR = Ao e (V.2l)
This corresponds to a penalty factor for statistical uncertaintywhere SN=SroOR.
on flR:
(See plots in Figure
~R5.)
(V.22)
(b) ° i unknown, flR known or unknown
In Section III it was shown that if flR is known, oR has noninformative
prior distribution (n'=a) and a sample of size n is given from the population
of R, the predictive distribution of the reduced variable R'=(R-flR)/S is t
with n degrees of freedom (S2 is the sample variance). Under the same condi-
tions, but with flR also unknown, it was found that-""
R' = (~)1./2. R -?Rn+-:i S
This fact allows one to study jointlyhas
the
t 1 predictive Bayesian distribution.n-two cases when ~ is known or unknown.
Let the reduced resistance R' be distributed like tv (i.e., v=n, or v=n-l),
with density:
r[(p-+-1)!2]
r(v!Z)• (V. 23)
Then the mean failure rate from earthquakes with (normalized) site intensity
between iNa and iNl is:
-' (V.24)
As iNa tends to ~ A. . . diverges for any finite v. It thereforef,v,J.N ,J.N
becomes important to establis~ a ~ight truncation point iNa for the resistance
distribution V (or indeed for the validity of the model as a whole), and to
(V.25)
74
exclude failure events caused by earthquake loads of smaller size. The "natural"
trunaction at MM intensity zero might be used for this purpose, but a higher
truncation point is often more appropriate. In fact, failures at very small
site intensities, say for 1<3, are due primarily to factors other than the
seismic load, for example, to very poor design, or to wrong selection of
materials, or to gross construction errors. In other cases, the simple know
ledge that the system survived previously applied loads (seismic or other)
guarantees truncation (or rapid decay) of the resistance density at low inten-
wity levels. The importance of iNa in the calculation of Af is apparent from
Figure 6, where the integrand in Equation (24):
i 2 )-(V-t-i.)/,2 ~i~ CiN , 17) [3N') = (1. + --;;- e - IV IV
is plotted versus iN for SN=l and for a set of V values. The reason for studying
this function is that it shows the relative contribution to the risk from
events with various site intensities. As v+oo g(iN,v, SN) approaches
exp {-~ i~ - SNiN} , Le. the integrand for normal :ttesistance densities (see
Equation IV.4).
Noticeable features of the function (25) are:
*
* g(',V,SN) has a relative maximum at
• -(1J-t-i)+-Vc V + i f-4 V (3/1/-II\! =
2 (3N
(as v+oo , this expression approaches -~N ),and a relative minimum at
(V.26a)
.ltv =
(V -I- 1) - -vc;; -t- i /- 4 V (31v2.
-2 frv
(V.26b)
When SN=1 (as in Figure 6), the relative maximum occurs at -1 for all v, andv+l
the relative minimum is at -v. For SN<2fV the function (25) decreases mono-
tonically with iN'
* lim g (iN,v,SN)= 00
iW-oo
lim g (iN,V,SN)=OiW 00
for all finite v;
for all v.
2
Af ; 00, _ 00, 00 = A 0 e ~w /2.
75
* For v+ro the relative maximum at -S is the absolute maximum, aboutN
which the function is symmetric.
For positive iN the functions g(iN,v, SN) are practically the same for all
V. A completely different situation is found at low levels of intensity,
particularly for small V (i.e., for large statistical uncertainty on the resist
ance parameters). In this case the risk contribution may even increase with
decreasing intensity, well within realistic ranges of iN values. In other words,
for small V the model suggests that if failure occurs at intensity iNo or higher,
it is most likely that the event was caused either by an earthquake with very
low site intensity (close to iNo)' or by an earthquake with site intensity close
to the value in Equation (26a). One should associate the former failure events
with systems having "very poor performance" (systems of this kind are infrequent,
but they rarely escape seismic failure, due to the high frequency of small size
shocks), and the latter failure events with rare, but highly destructive earth
quakes, having intensity levels close to (but smaller than) the mean resistance
of the system. Smaller risk is associated with earthquakes of intermediate
size, or with ground motions having site intensity larger than the mean
resistance of the system.
While iNO in Equation (24) depends mainly on the truncation of the resist
ance distribution, i NI depends on the site intensity upper bound. The curves in
Figure 6 show that the mean failure rate in Equation (24) is not sensitive to
iNI
, provided that truncation is above the mean resistance; instead, AF
may be
quite sensitive to iNO' particularly for small V. This is a qualitatively new
result in engineering seismic risk analysis, showing that combinations other
than "high demand-average resistance" may dominate the damage statistics, and
warning about the possible presence of an intensity range below the mean
resistance, with rather uniform contribution to the total risk.
Results from the numerical integration of Equation (24) (a resistance
density normalization factor [I-tv (iNO )] ·-1 was included in the calculations)
are qisplayed in Figures 7 through 12. In all cases the mean failure rate
Af;V,iNo,iNl is normalized with respect to the mean failure rate for no
statistical uncertainty (v+ro) and for an unbounded intensity range; i.e., with
respect to
For i NO=-8, the ratio Af ; v, - 8, 00
1\£;00,_00,00is plotted in Figure 7 for V=l,3,5,7,lO,20,
76
as a function of SN' The effect of statistical uncertainty on Af increases
dramatically with SN' and in all cases is non-negligible. It should be said,
however, that iNO=-8 is a rather conservative value for the lower bound. It
would result, for example, from a known mean resistance ~R=lO (MMI scale), from
an estimated standard deviation SR=l, and from a resistance truncation point at
MMI=2. For the same values of ~R and SR' but the lower truncation point moved
to 5, one should use the value i NO=-5. A second argument in favor af a higher
truncation point comes from the nonlinearity of the empirical log mean-damage
ratio as a function of intensity (See Section 11.2 and Figures 11.5 through
11.11). In the present "linear" resistance model (See Figure IV.l), the rapid
decrease of log MDR at low intensities can be approximately accounted for through
more severe truncations of the resistance distribution.Af .
Figures 8,9 and 10 contain plots of the ratio jV,ltVo'OO for i =-4(-1)-8Ar;oo,-oo,= NO
and for v=l (Figure 8), v=5 (Figure 9) and v=lO (Figure 10). These figures
confirm previous observations on the high sensitivity of the mean failure rate
to the lower truncation point, particularly for large values of SN'
The effects of varying the upper limit of integration in Equation(24) (this
limit coincides with the upper truncation point in the frequency-site intensity
law) are quantified in Figures 11 and 12 for v=5 and v=lO, respectively. In both
cases it is iNO=-8. The upper curves in these figures are for iNl=OO. It is
seen that for iNl>O the upper truncation has no appreciable effect on Af ; also,
for any given iNl' the effect of truncation decreases with SN' For example,for
i No=-8, the values of iNl in Table 1 are required to reduce the mean failure
rate A:f;V,-8,oo by a factor of 2.
SN \)=5 \)=10
0.8 -1.5 -1.0
1.0 -2.6 -1.4
1.2 -5.0 -1.9
1.4 -6.5 -2.7
1.6 -7.0 -4.3
Table V.l Values of i Nl such thatA f j ))) - 8, ~N:i.
2.
77
A •.f;V,l/V ,ll\l1-
Tables of the ratio A 0 are collected in Appendix A, for v=5,lO,20;£;00,-00,00
i No=-8(1)-3; i Nl=4(-1)iNo ; and SN=O.6(O.2)2.0.
V.3 UNCERTAINTY ON BOTH DEMAND jilin RESISTANCE PARA}IETERS
Consider the linear risk model
(V. 27)
and the normal resistance model:
R =/-1R + ER ERrvNCo;a-~) ,
rR J CT~ known or unknown.
(V.28)
For a seismic risk law-in the form (27) - 8.S opposed to the equivalent
it is reasona~le to assume that the seismic demand parameters, AiO and
independent of the resistance parameters, PR and 0i. In the remainder
present section two cases are considered: (a) AiO' SI and VR unknown,2
and (b) AiO' SI and OR unknown, VR known or unknown.
form (1)
SI' are
of the2
OR known;
2OR known
Let SN=SI'0R; i d=(io-OR)/0R (OR is an estimate of the mean of R), and
Aid=Aio have independent normal distribution:
is the mean rate of events with site intensity
0 )- (V.29)0
2.
vR /11
and that the parameter of
o2
o o-f3/V
o 0
2Implied by the model (29) is that R-N (OR,0 (l+l/n),
the frequency-site intensity law, written now:
A' - 'I - ~N i wIN - 1\0 e ,
-SI(OR-io)A =A' eo l.owhere
greater than OR' and
have the distribution (11) , independent of R. The mean failure rate is still
78
given by Equation (14), after replacing
~SN by ~SN (1+1/n)1/2
2 20SN by 0SN (l+l/n), and
(1+1/n)-1/2
The penalty factor for statistical uncertainty becomes:
, (V.30)
when all the parameters equal their mean values;
is given by Equation (4) , 2 0 2 (see alsoYAo with 01
Areplaced byld n 0 d
Figure 1);
YSN is given by Equation (6), with ~SN(1+1/n)1/2 in place of ~SN and
O~N(l+l/n) 2in place of 0SN (see also Figure 2);
Yid is given by Equation (15b) with the replacements above and, in
addition, i d (1+1/n)-1/2 instead if i d (see also Figure 4) ;
YJlR is given by Equation (22).
For n~ the factor (30) approaches the factor YAO,SN in Equation (15a). The only
partial factor in Equation (30) which does not depend on n is YA.• To exemplifyld
the dependence of the remaining partial factors - and so of Y - on n,AO,SN,JlR
consider the realistic case:
~SN=1.4; 0SN=0.2; i d=-5. The factors YSN"'Yid' YJlR and their product are
given in Table 2 for n=oo,10,5.
n YSN Yid YJlR YSN Yid YJlR
00 1.063 1.258 1 1.337
10 1.083 1.248 1.103 1.491
5 1.107 1.232 1.217 1.659
Table V.2
i d=-5.
Partial penalty factors in Equation (V.30); JlSN=1.4; 0SN=0.2 and
79
The increase ofYAo,~N,~Rwith decreasing n is due primarily to the factor ~R.
(b) 2Aio ' SI'OR unknown; ~R known or unknown
if ~R is known;
if ~R is unknown;
if ~R is known;
J if ~ is unknown;R
Consider now the case when the normalized resistance R' (see Section V.2b2for definition) has tv-distribution (23), as a result of uncertainty on OR ' and
possibly on ~R. Ao and SI are Bayesian random variables, independent of R' • Let:
SN=as in Equation (24);
{
Ci-rR)/S
~N == (1. + 1/:115:1./2 (i -JA-R) /s
. 5Ci o -/R) Islcl = ) -1../2---'\
l(1+ l./n) (io-rR)/S
\iJ"" Ai o •
The joint distribution of In Aid and SN is assumed to be normal, as in Equation
(29). If iNa is the lower truncation point of the resistance distribution and
i NI is the upper truncation point of the risk curve, the mean failure rate is:
Af· .,ltv. ,INo :1.
00 •
E [A i"JJI fl, (rN)J'C:r i ~ t-'+1)/~ - 11 (iN - i,1~LiN cLfN
- 00 lJl!o
where Cv is the constant in Equation (23); tv (.) is the CDF of tv;
.:J
Integration with respect to SN yields:
Af· ., 2No , IN 1-
i~ ) -(v +1)/2
(V.3l)
80
The penalty factor YAi ·6 0" (or YA· S II 0" ), which is the ratio betweend' N' R ld' N, R, ~
Af,iNo,iNl in Equation (31) and the limit mean failure rate:
lim 2on O"d,llSN'
v, i No and iNl:
Since the dependence on O"~ is
is convenient to tabulate the
This is done in Appendix B for the following values of the remaining parameters:
llSN=o.6,1.0,1.4,1.8
O"SN=0.1,0.2,0.3
i d =-3,-6
v =5,10
It is interesting to compare the exact penalty, Equation (32~ with the
approximation by partial factor multiplication; i.e. with YAid·YSN·Yid·YllR,O"R'
where
(Equation l5a);
is tabulated in Appendix A.
81
For example, using the following set of parameters' values:
Resistance parameters:
Load parameters: ic{ = - 6 ; /'f-v. i AI ==, Ive
i.4- ; 0(34-:= 0.2. ,i,y1. = 0 ;
- !:> ;
One finds, from Appendix B:
X - 2.699A icL' f1 '/R-' uR 2
i 2 J.lfN /2(meaning a mean failure rate: 2.699 ed/ • Aue ). From the partial factor
approximation one would find instead:
'YA · a-Jj.z . YicL = 1..'4-92 .l,L e , ,
1"(3,v 1. 063 . ~ () = 2~ lOr .,fU R :I
and
A more extensive comparison is shown in Tables 3 a,b,c,d, where the following
parameters' values are considered:
Table 3a: i,l == - 6 ;/'~== i.4 . art == 0.2 IN:l. =0 ))=5 ; i No := - B C1.) - "3~ , ;Iv
Table 3b: Same as Table 3a, except for il=-3;
Table 3c: Same as Table 3a, excep t for 0SN=O.3;
Table 3d: Same as Table 3b, except for 0SN=O.3.
Although generally conservative, the approximation by partial factors
gives penalties which are sometimes smaller than the exact values. One can
explain this as follows: in the approximation, the effect of SN unknown is
calculated under the assumption of normal resistance distribution, while in the
exact calculation the same distribution is of t-type (ts in Table 3). If
lidl is small (Tables 3b and 3d) and, at the same time, liNol is large, replacing
the normal distribution by the ts-distribution increases the risk from low
intensity earthquakes (See Figure 6), and introduces unconservatism in the
approximation. The approximation by partial factors is, instead, conservative
for larger lidl values, because in this case the sensitivity of the mean failure
rate to SN is smaller if the resistance distribution is t, than if the resistance
82
distribution is normal.
For some numerical evaluation of Af when both demand and resistance
parameters are unknown, see Section VI.
i No BY PARTIAL EXACTFACTORS
-8 11.74 8.16
-7 7.20 5.15
-6 4.78 3.61
-5 3.34 2.70
-4 2.39 2.07
-3 1.66 1.55
(a)
i No BY PARTIAL EXACTFACTORS
-8 22.09 9.56
-7 13.55 6.36
-6 8.99 4.80
-5 6.29 3.89
-4 4.49 3.22
-3 3.13 2.63
(c)
iN BY PARTIAL EXACT0
FACTORS
-8 7.97 9.54
-7 4.89 5.20
-6 3.24 3.22
-5 2.27 2.19
-4 1. 62 1.56
-3 1.13 1.10
(b)
i No BY PARTIAL EXACTFACTORS
-8 8.81 13.59
-7 5.40 6.24
-6 3.58 3.52
-5 2.51 2.31
-4 1. 79 1.64
-3 1.25 1.17
(d)
Table V.3 Penalty factors: YA" S ~ cr / cr~/2.J.d' N' R' R e
Parameters' values:
Table a: i d=-6; ~SN=1.4; crSN=0.2; i =0' v=5Ni. 'Table b: Same as Table a, except for i d=-3;
Table c: Same as Table a, except for crSN=Oo3;
Table d: Same as Table b, except for crSN=Oo3.
The exact values are from Appendix Bo
20
10
5
3
2
1
83
I I I I I I
f- -
/7
/'./
/ '"./
/"- / -
/v- -
c-- -
l-I-l-
I I I I I I
1 4 7 10
exp{ 0 ). JIn0
Figure V.l Penalty facto!" "'f>'o for statistical uncertainty
on the seismicity parameter \0'
Equation (V.4).
84
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Figure V.2 Penalty factor for statistical uncertainty on the
seismicity parameter BN• Equation (V.6)
~.~ \
J..11 ' o
85
t-'IM Intensity
Figure V.3 Seismic risk model for both Ao and BN unknown
86
0.2
0.1
0.6 1.2 1.8 2.4
llSN
2.41.81.20.6
1
1.5i d=-l i d=-2
1 1
0.5 0.5
0.6 0.6 2./{
50.3 5c pN=0.5 i
d=-3 i d=-4
0.4 0.2
L- 0.1
i d,
1 .....
0.1
0.5 0.5
0.6 0.6 2.4
i =-5 }d"'··6d
10 10
5 5
Figure V.~ Penalty factor Yid (Equation V.15b). BN, or SN
and unknown'0 '
(cant. )
100
,1.
i =-7d
87
i =-8100 t--_~~-t--_d__-+ --1
1 0 1--------=~::+-------I------1
0.1l'----t--...::...:.-=-I----J0.6 1.2 1.8 2.4 0.6 1.2 1.8 2.4
88
2.0
1.8~--------+---------+------I------I
1.6~--------+---------+---I---~-----I
1.4~--------+------~~-~----~:"-'----I
1. -;.-
1.0'--_--'-__..........__'--_----I.__....1-__.L.-_-..L__....1-_~
0.6 1.2 1.8 2.4
Figu~~ V.S Penalty factor I~R (Equation V.22) for unknown
mean resistance. n=available sample size (for noninformative
prior distribution of ~R)'
89
I,,"- 'J-?OO
I (normal resistance)
I
\~! =8
10
1 0-1t-----+------Ir--I----+-----+----.!"--+---'\----I..L ~ :
1 0~ 2~____'_ ____'.A-----l_....LJ._ __l_ __l____l_ ___L_ ___L_ ___L....AI.......J_--1
.-1
IIZ
'•..!..}
rl±~.
-..,/
I
(". iT:>·C'l ~
....,....~
-8 -6 -4 -2 o 2 4
Figure V.6 2uR
unknown; WR
known or unknown. Plots of the
integrand in Equation (V.24) for SN=1 and v=lt4t8t20too.
" 03
.L
3I
6
401.-< 2
':," 10
~
:.>
10
1
0.6 0.8 1.2
90
1.6 2.0 2.4
Figure V.7 Penalty factor for 0~ unknown and WR known or
unknown. Resistance truncation point iN =-8.o
91
tl
, truncated
Linear seismic risk model,no truncation
RESISTANCE TRUNCATION, iNo
".L
103 t=--------j--------i-------JL------,JL+---l
a3I
3102
'H<-.;
........;
0;?;
'r~~
,..;
'i-j
.-<
10
0.6 1.0 1.4 1.8 2.2
Figure V. 8 Penalty factor for c; unknown and ]J R known or
unknown; v =1; vari.able lower resistance truncation
point.
92
RESISTANCE TVUNCATION,i~
-8
1
102 t----+--------If----------+--
103 ~--r---r---r----,~---'-----"''''-----'-----'----
"I
8
0.6 0.8 1.2 1.6 2.0 2.4
Figure V~1.2Penalty factor for CR unknowll and flR known or
unknown. v=5; variable lower resistance truncation
point.
93
2.42.0
iN =-8a -7
-6
-5
-4
1.61.2
10g
gI
8.~
lH..-<
-....g
0z1"rl
00.6 0.8....;
lH.-<
Figure V.IO 2Penalty factor for OR unknown and W
Rknown or
unknown. v=lO; variable lower resistance truncation point.
94
6 ~ ~OAD TRUNCATION,iN7 .. f----------------""-'-±l
tRESISTANCE TRUNCATION
3
310
10
i N1= :x..
d"2
I"8
lH,-«1---.
,-j,~
·,-r'"(X)
I"1Il
4-l<
10-1
0.6 1.0 1.4 1.8 2.2
Figure V.1l 2Penalty factor for OR unknown and ~R kr.own or unknown.
v=5; variable upper bound on site intensity, i N1 "
95
f-- LOADTRUNCATION,
iNl
-7
-·3
-4
-6
-s
~.
. 0- 3.L LL.__--L ...L ..l.-__---I --l.. ....I-__.......JI--__-..l..__.....
10
10- 2 t----:;~---f-----J<-+------+_-----___+------_+_-__1
'3.8I
C)rJ
c<4-l 10-1 I---~~---,A-~'------'-"---,~-+-~----'~
-r-J
0.6 1.0 1.4 1.8 2.2
Figure V.l2 Penalty factor for 0~ unknown and WR known or unknown.
v=lO; variable upper bound on site intensity, iNl
.
96
VI. PARAMETERS SELECTION AND RISK EVALUATION
The information summarized in Section II is used now to select the para
meters of the seismic risk model (or their Bayesian distribution).As an
application example, the seismic risk of typical nuclear power plants located
in Eastern United States regions is calculated, and compared with approximations
from Equation (1.4).
VI.I SELECTION OF RESISTANCE PARAMETERS
The resistance parameters of the probabilistic Gaussian model in Figure
IV.I are ~R and aR. If a statistical Gaussian model is used instead (see
Section V.I), the following information must be provided:
n, OR and aR if ~R is unknown and aR is known;
V, 0R,SR and i No if aR
is unknown and ~R is known or unknown
In consideration of the limited information available on resistance parameters,
the last assumption - aR and ~R unknown - seems to be the most realistic one.
Estimates of aR are available for ordinary civil and industrial constructions
(aR=s~l and estimates of SD are given in Table 11.5), in which case SR varies
typically between 0.50 and 0.65. Higher values are found using data from
Newmark (1974) and from Vanmarcke (1971). From Newmark's data one calculates
SR=0.75 for nuclear reactor structures and ordinary civil constructions and
SR=0.86 for nuclear equipment. These values refer to seismic demand and
resistance expressed originally in units of log peak ground acceleration, and
then converted to MMI, through the solid line relationship in Figure (11.4).
In the same sense, the data in Vanmarcke (1971) suggest SR values between 0.65
and 0.70 for ordinary civil constructions. Reasonable values of SR might be:
(b) OR
[
(0065,the range
(0070,
0075)
0085)
97
for nuclear reactor structures;
for nuclear reactor equipment.(VIol)
For ordinary buildings OR can be estimated, for example, as the intensity
at which the appropriate line in Figure 11010 reaches the critical MDR value,
dfo For reactor systems and components it was concluded by the USAEC Nuclear
Reactor Safety Study WASH-1400 (draft report) that the probability of failure
under the Safe Shutdown Earthquake (MM intensity i SSE) is in the range 10-1
to 10-2 (*)0 For normal resistance distribution this implies an estimated mean
value of R:
() in the rangeI-lR {
(iSSE+0.9' iSSE+l.62),
(iSSE+l.09,iSSE+l.97) ,
for
for(VI02)
For most nuclear power plants either in operation or under construction in the
Eastern United States the peak ground acceleration for the SSE, aSSE ' is about
0017go This value corresponds to a Modified Mercalli intensity i SSE of
approximately 8 (see Figure 11.4) and to the following ranges for;4R:
A { (8.9,9.6) for SR=0.70 , (a)1\ in the range (VI. 3)
(9.1,10) for SR=0085 (b)
(c) \!
The"confidence parameter" \! is not easy to establish because the information
on R is rarely in the form of a statistical sample. It is suggested that values
in the range 5 to 10 (corresponding to "equivalent sample sizes" from 6 to 11)
(*)Newmark (1974) suggested failure probabilities for nucle~4reactor equipment under the design earthquake of the order 10-2 to 10 ,or smaller
98
may be appropriate.
For a set of independent N(O;l) variables Yl' ..• 'Yn the statistic n %has Student's t distribution with v=(n-l) degrees of freedom. This means that
if OR is chosen to be the center value of the intervals (VI.3), the same
intervals contain ~R at the following confidence levels:
0.74 for v=(n-l)=5
0.88 for v=(n-l)=lO
Arguments of this kind can be used to select an appropriate value (or range
of values) for v.
This is another parameter which is difficult to establish with high
confidence. If one defines "seismic failures" to be those triggered by ground
motions of intensity VI or larger, then i No should be given the following values
(iNo~(6-0R)/SR):
(-5.1, -4.1) for SR=0.70 and (a)
OR in the range (3a) ; (VI. 4)i
Noin the range
(-4.7,-3.6) for SR=0.85 and (b)
OR in the range (3b) •
If the threshold intensity is lowered to V, the range (4a) becomes (-6.5,-5.5),
and the rnage (4b) becomes (-5.9,-4.8).
VI.2 SELECTION OF SEISMIC DEMAND PARA}ffiTERS
The exact selection of seismic demand parameters can be done only with
reference to a specific grographical location. However, using regional seismic
information and typical attenuation laws, ranges of parameters' values can
be estimated, sometimes over large areas.
For complete characterization, the linear frequency-site intensity law
(IV.3b) requires knowledge of Ao (mean rate of events with site intensity
99
greater than OR) and SN if the model is probabilistic; of i d , ~d=E[ln Aid] ,
~SN' °d=0ln Aid ' and 0SN if the model is statistical. For truncated linear
risk laws, the upper bound site intensity i Nl must also be given. The analysis
in Section V did not allow for statistical uncertain~y on i Nl • If the intensity
upper bound is larger than the mean resistance, the effect of this uncertainty
is negligible; if instead the upper bound is smaller than OR' the mean failure
rate calculated in previous sections and tabulated in the appendices for
different values of i Nl can be used to establish the effect of i Nl uncertainty.
The following ranges of seismic demand parameters are consistent with recent
risk calculations for Massachusetts (Cornell and Merz, 1974; Tong et aI, 1975).
With obvious caution, the same ranges can be considered typical for many regions
in the Eastern states. Figures 1,2 and 3 (solid lines) are from Tong et al
(1975). They give the annual seismic risk in MMI at five different sites in
Massachusetts, under different assumptions ~n the geometry of the seismic sources.
In all cases the maximum epicentral intensity 10
was assumed not to exceed 8.7.
The site intensity i at which Ai and SI can be considered independent of
one another is approximately V; see Figures VI.l,2,3, and Figure 11.1. From
the ranges of mean resistance values (3), the normalized intensity id~(i-OR)/SR
is then:
i d in the range{
(-6.5,-5.5),
(-5.9,-4.8),
for SR=0.70 and OR
in the range (3a);
for SR=0.85 and OR
in the range (3b)
(a)
(VI.5)(b)
At all sites within Massachusetts analyzed by Cornell and Merz (1974) and
by Tong et al (1975), and under all the assumptions made by the same authors
about the seismic sources and the regional seismic parameters, the mean annual
rate of seismic events with site intensity larger than V was found between
0.7 x 10-2 and 2.3 x 10-2 • One may ther<cfore assume:
100
(VI. 6)
and ad values between 0.1 and 0.5.
In Section IV (see Figure IV.6 and Table IV.2) SN=SI'aR was found to
have values in the range 0.90 to 1.60. Linearization of the curves in Figures
1,2 and 3 gives SI values of about 1.45 to 2.30, corresponding to
{
(1. 0'.1. 6) ,SN in the range
(1.2,2.0),
(a)
(b)
(VI. 7)
Appropriate values of ~SN and aSN
for Massachusetts might then be:
The upper bound site intensity varies from region to region and, within
each seismic region, from site to site. For Massachusetts, values between 7.5
and 8.7 seem reasonable (see Figures VI.l,2,3 and Figure 11.1). If the
resistance parameters are in the ranges (3), these values correspond to
i Nl in the range
5(-3,0),
1(-3,-0.5),
(VI.8)
Upper bounds cannot be established with certainty; indeed, according to a few
seismologists (see, e.g., Chinnery and Rogers, 1973), epicentral MM intensities
as high as X are possible in Massachusetts. An appropriate practical upper
bound for site intensity might be iNl=-l.
Althbugh not treated explicitly here, statistical uncertainty on i Nl can
be incorporated (approximately) into the analysis by assigning (Bayesian)
probabilities to a discrete set of i Nl values, and by weighting the associated
101
risks through the same probabilities. The tables in Appendices A and B would
be helpful for this type of analysis.
VI.3 MEAN FAILURE RATE CALCULATIONS
From the preceeding discussion, the following "best" parameters' estimates
are suggested:
* For nuclear power plant resistance: ).lR=9.5;
V =10;
SR=0.75;
iN =-5'o '
(VI.9)
* For seismic risk at a Massachusetts site (see dashed lines
in Figures 1,2,3):
i =-6 .d '
).ld --2e =1. 3 x 10
i =-1Nl
(VLlO)
For an untruncated linear risk function, ).lIn Ao
satisfies:
e)Ad . C': r/1 .icL = o. .z:3 " 10 - 5",
and the mean annual failure rate is:
Using
and
Af =0.29 x 10-5 x 2.66 x 1.08 x 1.468 = 1.23 x 10-5 (VI. 11)
The values (9),(10) correspond to a probability of failure 0.046 for an event-4with site intensity 8, and to a mean annual rate 0.48 x 10 of exceeding the
same intensity. Using the approximation (1.4) one finds Af~0.22 x 10-5 ,
which is about 5.6 times smaller than the estimate (11).
102
Conservative estimate of Af
Consider the following "pessimistic"parameter values:
i d=-6
°d=0.5
Os =0.3N
SR=O. 7.'5
i No=-5
lld -2e =2.0xlO
llSN=l. 40
iNl=4
(VI. 12)
The associated mean annual failure rate is:
-5 -5Af =0.446 x 10 x 2.66 x 1.133 x 4.404 = 5.92 x 10 (VIo13)
The difference between the estimates (11) and (13) is due mainly to lowering
V and to increasing 0SN. Using iNl=-l, and the values (12) for the remaining
parameters, one finds Af =3.97 x 10-5
For an upper bound site intensity 8, for which i Nl=-2, the mean failure
rates (11) and (13) become: Af=0.75 x 10-5 , and Af=2.71 x 10-5 , respectively.
Changing i No by ±l produces a change of about 10% in the estimate (11), and
a change of about 20% in the estimate (13).
Other sensitivity analyses are easily made with the aid of the tables in
Appendices A and B.
The preceeding calculations refer to the mean annual rate of accident
initiation in a specific mode (e.g. by break of a pipe in the primary coolant
system). Many different events may trigger an accident sequence, and eventually
lead to core melt and to radioactive releases. The probabilistic analysis of
all such sequences is complicated by two factors: (i) the statistical
correlation between different failure events (which, should they occur, would be
caused by the same ground motion), and (ii) the redundancy of nuclear reactor
systems and safety devices. The inclusion of these features in the seismic
risk analyses of complex systems is possible within the methodology proposed
and illustrated in this study. In fact, the probability (IV.l) - interpreted
103
as the resistance CDF - should account implicitly for all possible failure modes.
This is the case, for example, when Pf(I) is estimated from historical records
of seismic damage, such as those reviewed in Section 11.2. Instead, the
calculation of Pf(I) for the whole system is not an easy task when starting
from the failure probabilities of subsystems or components, as is usually
the case in structural reliability theory. The explicit consideration of this
problem is left, however, for future efforts.
')
lO-L
~
f-tHV)
ZH0~WUxCt.1
p::;,..;
0 10- 3~~H
:;;j ?-';::J bYHr.LI rjJ
Z~ ~C) b
Z;,,: ;.....(U");...;p::;
....:l~;::Jzz<4
lO~4
IV V VI
104
VII VIII IX
MINIMU}1 RESISTANCE INTENSITY UPPER BOUND
HM INTENSITY, I
Figure VI.l Solid Lines: Annual probability of equaling or exceeding intensity
I at five sites in Massachusetts (after Tong et ai, 1975). Dashed line:
truncated linear approximation
MINIMUM RESISTANCE
lC2
>-<l-<H(f)
Z~HZ}-1
.~t-<H(j)
0;;Z;HQ;'1~ . ,
10-3uxwp::0
~)
ZH.....l<t;;:JCYW
::G0
::.::r.IlHrt:
H -4<t; 10::.J;z;Z<1;
IV V
105
VI VII VIII11M INTENSITY, I
INTENSITY UPPER BOUND
IX
Figure VI.2 Solid lines: Annual probability of equaling or exceeding
intensity I at five sites in Massachusetts (after Tong et al, 1975)
Dashed lines: truncated linear approximation
106
IXVII VIIIINTENSITY ,I
INTENSITY UPPER BOUND
VIMM
V
MINIMUM RESISTANCE
10-5L- ....I- I--......... --'- "--......L_..L......."--..!
IV
_'J10 ...
;,,-.f:-lHtilZ~
f.-1Zc...,
"-lt-<'4til
~ '"'-'Z>--J~>x.1l.J;<
u -1:x 10 -}i.]
p(.()
~)
Z'-i.....:i<t;~)
0'~I
Lt..;0
~tilH
~
~ 10-4~:'.;
3--.
Figure VI.3 Solid lines: Annual probability of equaling or exceeding
intensity I at five sites in Massachusetts (after Tong et aI, 1975) Dashed
lines: truncated linear approximation
107
REFERENCES
Albee, A.L. and Smith, J .L. (1967), "Geologic Site Criteria for NuclearPower Plant Loaction," Trans. ,Society of Mining Engineers,Dec. 1967, 430-434.
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Algermissen, S.T. et al (1969): "Studies in Seismicity and EarthquakeStatistics, Appx B,C, and GS," U.S. Dept. of Commerce, Coastand Geodetic Survey, Rockville, Md.
Algermissen, S.T. (1969): "Seismic Risk Studies in the U.S.," 4th WorldConference on Earthquake Engineering, Santiago, Chile.
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Bollinger, G.A. (1973)" "Seismicity of the Southeastern United States,"Bull. Seism. Soc. Am., Vol.63, 1785-1808.
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108
Cornell,C.A. and Merz, H.A. (1974): "Seismic Risk Analysis of Boston,"ASCE National Structural Engineering Meeting, Cincinnati, Ohio,April 1974.
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109
Hong,S.-T. and Reed,J.W. (1972): "1965 Puget Sound, Washington, EarthquakeTall Building Damage Review," Seismic Design Decision Analysis,Internal Study Report No.23, Dept. of Civil Eng., M.I.T.
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no
Nutt1i ,O.W. (1974): "Magnit:ude Recurrence Relation for Central MississippiValley Earthquakes," Dept. of Earth and Atmospheric Sciences,Saint Louis University, St. Louis, Mo.
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111
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a,b
(M)DR
I,i
io
M,m
R'
Y,y
112
MOST FREQUENTLY-USED SYMBOLS
constants in the frequency-magnitude law, Eq.(II.2)
(a is also used for peak ground acceleration, depending on context)
constants in the epicentral intensity-magnitude law, Eq.(II.6)
parameters of the linear intensity-mean damage ratio model,Equations (11.15) and (11.16).
constants in the acceleration-magnitude-distance relation (11.14)
damage
(mean) damage ratio
critical level of damage or of damage ratio
MM site intensity
MM epicentral intensity
same as I,i; normalized
same as i , normalizedo
MM intensity, such that Aio is independent of SN; see Figure V.3
MM intensity upper bound
Richter's magnitude
magnitude upper bound
failure probability
seismic resistance (also used for epicentral or focal distance;see context)
normalized seismic resistance; see Section V.2.b
measure of seismic intensity
design seismic intensity
b In lO=constant in the frequency-magnitude law
decay parameters of the epicentral and site intensity distributioTIS j
Eqs.(II.7) and (11.13)
same as SI' normalized
penalty factor for uncertain resistance
211,O
2O,s
113
penalty factor for 8l , ••• ,8m unknown; see Section V
standard normal CDF and PDF
mean rate of events with intensity larger than the designintensity
mean failure rate
approximate mean failure rate; see Eq. (IV.5)
mean rate of events with normalized I:1M intensity larger than i d
mean rate of events with site intensity larger than the expectedresistance (see Section IV.2 for more precise definition in thecase of nonlinear risk models)
mean, variance
2unbiased estimates of 11 and a
vector of unknown parameters; see Section III
114
APPENDIX A
PENALTY FACTORS FOR UNKNOWN OR AND KNOWN OR UNKNOWN llR
Unknown OR and known or unknown llR; tables of the ratio
Af
. .;\),lNo,lNl
Af.OO _00 00" ,
see Equation (V.24). The tables are for \)=5,10,20; SN=0.6(0.2)2.0;
i =-8(1)-3-No '
** » = ~
115
r",=C.60 **
L.Aro
[,N. -8.e -7.0 -6.0 -5.0 -4.0 -3.0:1
4.0 1.155 1.139 1. 121 1.(98 1.C64 I.CIC3.0 1 .154 1 .138 1.110 1.('97 1.e63 1.C092 .. C I .146 1.131 1. 113 1.e9C 1. C56 1 • a011 .. 0 1.IJ99 1.084 1.0t5 I.C42 1.COE C.9530.0 0.R95 O.P,79 0.R61 O.f38 0.803 C.74t
-Ie: ::.542 C.5?7 (.5'18 0.L,84 U.449 0.388
-2.0 ".2R7 0.271 C.253 C.22<1 0.192 0.129-3.0 :.1';(1 0.144 0.116 O. 101 0.064 o.c-4.C (.C96 e.cec C.062 (.(37 o.c 0.0-5 .. 8 n •D 'J? 0.043 C.025 c.c G.G c.e-6.0 0.034 J.OIQ c.o 0.0 o.e c.e-LC C.016 c.c 0.0 o.c 0.0 0.0
** )} = ? ~N =0.80 **L.No
/.-!vi. -8.'! -7.0 -6.r) -5.0 -4.0 -:3. C
4. C 1.-~9'J 1. ~-H 1. ? 71 1.708 1.133 1.0323.C 1. V39 1.329 1.271 1. 20 E 1.133 1.G312.e 1.380 1 .3;:> ') 1.267 1.204 1.12<7 1. C2 7I.e 1.354 1.294 I .235 1.173 1.097 0.995---, " 1.1 gJ 1. 13 C 1.071 1.COE C.SJ3 G.829V.l!
-LO ').851 0.790 0.732 0.t6e 0.591 C.484
-2.0 C.554 0.494 (.435 C.371 0.293 0.183-3.0 0.373 0.313 J. 254 C.Ise 0.112 C.G-4.0 0.262 0. 7 02 0.143 0.G7g C.G c.c-5~~ C. 1 R4 C. 123 C. 0 64 ':: •c o.c o.c-6.G () .120 0.059 C.IJ c.c o.c c.e-7.C r.06,) 0.0 0.0 O.C o.c C.C
** y = 5
116
l-NO
i,yj. -8.0 -7.0 -6.C - 5.0 -4.0 -3.0
4.C 1.<)5'1 1.733 1.553 1. 393 1.236 1.0593 n 1.960 1.733 1.553 1.393 1.23l: 1.C59. ~
2.0 1.958 1.731 1.550 1.391 1.234 1.C57I.e 1.937 1. 711 I.S'J 1.371 1.213 1.037o.c 1. B11 1. 584 1.404 1.744 1.086 C.9O P
-1.0 1.496 1.26<; 1 .028 0.928 0.76<; C.SEe
-2.0 1.163 C.936 C.755 0.595 0.435 0.25C-3.0 0.916 0.690 C.508 C.348 0.187 C.G-4.C 0.730 0.503 0.322 0.161 c.c c.e- Cj. C ':.569 C.342 c. }61 o. (, 0.0 c.o-6.G C.4C~ C.181 c.o c.c c.c c.c-7. a 0.727 0.0 0.0 'o.c c.e c.c
(1=1.20 **
tNo
'--Nt -~.c -7.C -t.8 -5.0 -4.0 -3.0
4.C 3./+ 19 2.619 2.083 1.b96 1 .382 I.C913.C "3.43'] 2.6 J 9 7.083 1.696 1.382 1.091.., ') 3.43B ('.tlE 2.GP2 1.t95 1·.3 e 1 l.C9C( .....1- r\ 3.475 2.605 2.070 1.(;83 1.36E 1.C778.0 3.331 2.511 1 • q 75 1.58B 1. 273 0.981
-l.0 3. J.'ig 2.22S 1.693 1.306 0.99C 0.695
-2.0 2.6119 1 .869 ] .333 o. gft'S O.t2E C.33C-3.e 7.364 1.544 1 .008 0.619 0.302 0.0-4.C 2.064 1. 244 C.707 0.31S C.C C.C-5.C 1.746 0.926 0.32] O.C O.C C.C-6. e 1.357 0.3,1 0.0 O.c J.G O.C-1. ': C.q2J ~ r C• ,; C r o.C c.e....... '..... • v
** )} = 5
117
~ =1.4C **Iv
{,l\Io
[,Ni.-8.C -7.e -6.0 -5.C -4.0 -3.0
4.C 7.476 4.617 3.0R7 2. 183 1. 581 1. 1233.C 7.476 4.617 3.0R? 2. 183 1.581 1.1232.0 7.476 4.l:1c 3.086 2. 183 1.581 1.1231 .0 7.469 4.60g 3.079 2.175 1. 573 1.115O.G 7.401 4.54 I 3.0 II 2.107 1.505 1.C47
-1oC' 7.158 ':'.29£ 2.763 1.864 1.261 0.80C
-2.e 6.783 3.923 2.393 1. 488 0.884 C.ltlS-3.C 6.370 3.589 1.979 1.074 0.469 c.c-4.C 5.ge3 3. C4 3 1. 512 0.607 o.e c.e-5.0 5.297 2.437 C.906 o.e o.c c.e-G.C 4.3'12 1. 531 0.0 " " 0.0 C r'.j • I...... ....-7.e 2.861 r (' C.O C.C o.e 0.0" .-
**)./= 5
&M0
L,v::i -P. J -7.'": -6. :~i -~ .0 -4.0 -~.c
4.C IP.3C9 9. 191 4.9R5 2.<;52 1. 245 1.15"330 C 18. R09 9.191 4 .f) es 2.952 1. e4:: 1. 1532. C le.ROg 9.191 4.9d5 2.'152 1. 84 5 1.153I.C P.804 '1.187 ~.9FlC 2.<]48 1. F4 C 1.14 P0.0 1~.75'3 9.14 C 4.934 2.901 1.7'13 1. 1 C1
-I • C 18.555 8.938 4.711 2.698 1.590 C.R9l:
-2oL l Q .179 8.561 4.3~4 2. 321 1. 212 0.513-3.0 17.671 8.055 3.r/~8 1.814 0.703 c.c-4." 16.973 7.355 3. ]1,8 1.113 o.e o.c- 5. (1 15.362 6.244 2.036 c.e c.c C.:-(;o(j 1J.P27 4.209 0.0 o.e o.c c.c-7.( S.61g c.o c.o o• c., 0.0 o.e
** )} = 5
118
11 = 1.8 C **N
l-No
LN~-8.0 -7.0 -6.0 -5.0 -4.0 -3.0
4.0 50.866 1 eLt;94 e.S46 40 142 2.17S 1.175~.O 50.866 19.694 R.546 4. 142 2.17'1 1.175l.C 5C.~66 19.614 8.546 4 .142 2.179 1.1751.0 SC.Re3 IS.t;91 8.544 4.14C 2.177 1.172o. c! 50.P32 19.66C P.512 4. 1C~ 2. 14 t 1. 141
-I.e sr;.670 19.498 £.350 3.<)46 l.gS3 O.97t
-1.0 5C.305 19.133 7.9 r35 3. 591 1. t 1 t C.6Ct-3.0 49.708 18.536 7.388 ;?982 1. CIt c.c-4.C 43.697 17.525 6.376 ·1.969 o.c c:.c
- r 40.731 15.552 4.4C9 c.c ~ r C.t)-:>.\... Uo~
-6.0 42.325 1 L 152 c.o c.c c.c c.c-7.0 31.176 0.0 c.o 0.0 o.c o.e
** )J = ') ~=2.CO **
tIVo
"-IV1- -8.0 -7.r: -6.0 - ';. C -4.0 -3.0
4. ( ]4(:.°82 43.606 1') .134 ') ('- Q 2.588 1.18L:• '-J j '.;
3.e 14C.922 4~.tCt 15.134 5.'::;38 2.588 1.1842 (. 14·::;.q82 43.cOt 15.133 5.93E 2.?EE } • 1 e 4• c
] • C 14C.9l:lQ 43.60:; 15.132 5.q37 2.58t 1.183c. (, 14C.9tO 43.~e5 15.112 5.917 2.::66 1.162
-1.G 140.835 43.46C 14.987 5.792 2.441 1. C3t
-2.0 14C.495 43.12C }l,.646 5.451 2.C99 0.690-3.e 13<LP.15 42.43<; 1~.9U; 4.7t9 1. 415 c.c-4. C" 138.407 41 .031 12.557 3.358 o.c c.c-5.C 135.('54 37.678 9.702 o• ,) o r, o.c.v
-6.0, 125.858 2E.48C C.G r (' C.O O.cv • ~
-7. r q7.3R4 0.0 C.O O.C O.C C. C
**V=10
119
&No
llVj.-8.0 -7.0 -6.G -5.C -4.0 -3.0
4.( 1 .054 1. C53 ] .051 1.047 1.03c 1.0073.C 1.053 1. C5 2 1.05C 1. C46 1.036 1.0072.0 1.047 1.046 1.044 1.040 1.C3C 1. CC 11.0 0.998 C.997 0.995 0.99] 0.981 0.9510.":- C.7A6 C.185 :'.:.783 C.179 C.76? C.73?
-lor' O.41Q 0.4113 C.41t 0.412 C.401 0.36<;
-70C C.l')B G.157 .0.155 O. 151 o.14C 0.106-300 C.051 C.C52 c.ose C.C46 0.C34 C.C-40 C C.D19 O.ClS 0.016 0.012 o.c c.c-50C ':.c07 C.C06 C.004 o.e o.c o.c-60C. C.2C3 C.C02 C.O c.c o.c c.o-7.0 C.OOI c.c c.o c.e c.e. c.c
** v= 10 ~N =0. eo **
LlVo
IN:i -p.e -7.e -6. :: -5.0 -4.0 - 3.0
4J ] • 1 15 lolL' 1 • ] ')6 1.CQ5 1.072 1. C1 P3. ( 1 • ] 15 1. 111 I .105 I • C9 5 1.::; 71 I .01 E2.e 1. I 11 1. 1 CE I.IS? 1. C9 1 I.C68 1. C141 • (, 1.079 1.C75 I •.C7 C I.C5S 1.C35 C.<;8100C! 0.909 0.905 C.9JO :::I.R89 0.B65 G.8IC
-I.e C.S55 C.552 C.546 C.535 0.511 0.454
-200 G.?52 0.249 0.243 0.232 0.20E C. 14 S-3oe 0.IC4 C.IOC 0.095 0.084 0.059 c.o-4.C C.045 C. 041 C..035 C.C24 o.c c.c-SoC '1.020 0.017 0.011 o.c 0.( c.c-6.( c. ceq C.C06 0.0 c.e 0.0 0.0-7.r:' ~.C~4 c.e c.o e.c O.G 0.0
120
ev=l.CO **-
-3.~£-/110
---tAl:!.
-8 .. e --70 0 -6 .. 0 -5.0 -4 .. 0-
4.0 L229 1.216 1,,198 L 171 L.123 LC3C3 .. C 1,,229 L216 1 .. 198 1 .. 171 1..123 1 .. 03C2eC L227 10 21lj 10197 1" 169 1..121 1,,0281 • U 1 .. 201 1,,194 1..116 1" 14 e 1" 1CC leC07O.. Ci 1 .. 075 10062 1.,044 1"Cl1 0,,969 0 .. 815
-1.(' 0.741 C 734 (,,116 0 .. 689 0,,640 0.,544
-2" C' .] e4 01 0,394 0 .. 377 0.3419 C.. 3CC Ce 203- 3 e ( C .. 2 G6 0,193 C .. 175 0 .. 14~ G.. 098 C.C-4.( C.le8 C,,(95 C.017 G e C4g O.C 0.0-5.0 0.059 0" 045 ( .. 028 Cee Gee Cec-6eO 0.031 0.e18 CeO 0 .. 0 O.. C G"C-7.C 0.013 Goe c..O e.c O.. C 0 .. 0
**))=10 ~ =1 .. 2C **AI
[.~
i lV1. -9 Ct fJ -1.0 -6.0 -5,,0 -4.0 - 3. a
4.C 1.454 1.402 le356 1.29C 1"lY4 1 • 04"3i.e 1.454 le40~ 1.355 le28CJ 1. 194 1" CLd2,,0 10454 1.407 1.355 1 ,,289 L193 1.C42LC 1.4/d 1.394 1 .. 342 1.276 1 .. 181 1.0290.0 1.343 1.296 1 .. 244 1. 11 E 10 C83 C.S3C
-LO 1.049 1.002 0 .. 950 00884 0 .. 788 0 .. 634
-2.0 Coh83 C.t3t C.. il)R4 C.. S1? C.422 0 .. 266-300 0 .. 419 0.372 0.320 0,,254 0.157 C.C-4.C C0262 0.215 C.163 0.096 O.. C C.. O-5.0 C.ll:5 C.118 C.1)66 C .. C G.C 0 .. 0-6.0 0.099 0.052 C.O c.e c.e c.e-7.e 0.047 0.0 e.1) aoo 0.0 0.0
** V=10
121
~ =1.40 **IV
Llvo
LN-8.0 -7.0 -6.0 -5.0 -4.0 -3.0
1..
4.0 1.933 1. 771 1.623 1 .. 471 1.290 1.0533.0 1.933 1.771 1.623 1.471 1.29C 1.C537.0 1.932 1.770 1.623 1.470 1.290 1.0531.0 1.925 1.762 1.615 1.462 1.282 1.0450.0 1.854 1.692 Ie 545 .1.392 1.212 0.974
-1.0 ] .601 1.439 1.791 1.1'39 0.958 0.1lf~
-2.0 1.220 1.05£1 (;.910 G.7S? C.. 51l:. C.33t-3.0 0.887 0.724 0.577 0.424 0.243 O.C-4.2 0.644 8.482 0.335 0.182 0.0 0.0-5.0 0.463 C.30e C.153 c.e o.~ 0.0-6.0 0.310 0.148 c.o o.c o.c o.c-7.0 0.162 0.0 0.0 0.0 0.0 c.o
** V=10
ilV.0
lNi.. -p.e -7.0 -6.0 - 5.0 -4.0 -3.0
4.C 3.025 2.494 20083 1. 143 1.414 1.0593.0 3.025 2.484 70083 1. 743 1.414 1.0592.0 3.025 2.434 20022 1.742 I. 414 I.C5S1.0 3.021 2.480 20078 1 .. 738 1.409 1.054o.c 2.912 2.431 2,,030 1.69C 1 .. 361 i.Oos
-1.0 2 .. 761 2 .. 221 Iv R 19 1.479 I.15e 0.793
-2.C 7.379 1.838 1,,437 1. C96 0.767 0.409-3.e 1.973 1.433 L031 0.t:91 0.361 c.c-4.0 1 .613 1.072 0,,671 0.330 o.c c.e-5.C 1. 283 J.742 0.340 o.c 0.0 0.0-6.0 C.943 C.4C2 c.o c.c O.C c.c-7.e 0.541 0.0 c.o 0.0 o.c c.e
122
'--No
LN 1.-Q.O -7 .. 0 -6.0 -5 .. 0 -4.0 -:3.0
4.e 5.668 3.93C 2.874 2.145 1.567 1.0563.0 5.t68 3 .. 'nc 2.874 2. 145 1.567 I.C562.G S,.668 3.93C 2.874 2. 145 1.567 1.0561.( 5.665 3.927 2.B72 2.142 1.565 1.(54o.c 5.633 3.895 2.840
j2.110 1.533 1.021
-I.e 5.464 3.726 2.671 1. S4 1 1.363 C.851
-2.C 5.095 3.357 2.301 1.511 0.994 0.48C-3.C 4.618 2.E8C 1 .. 825 L C95 C.517 C.G-4.0 4.10? 2.364 1.309 . 0.579 o.c c.e-s.c 3.524 1.786 0.730 0.0 0.0 O.G-6.0 2.7g4 1.C55 c.o c.c c.o 0.0-7.0 1 .7313 0.0 c.o c.c c.c c.c
**))=10
'--Ala -[, J\I:i -8.C -7.C -6.0 -5.C -4.0 -3.0
4 " 17.793 6.908 4.236 2.727 1.75C L(43.v
"3. C 17.793 6.90B 4.236 2.727 1.7SC 1.0432.0 12.293 t.soe 4.236 2.727 1.750 1.C431 • Cl 12.2ql 6.906 4.2;5 2.725 1.74S 1. (41, " 12.2-fl 6.8'36 4.214 2.705 1.728 1.C2CU.\.J
-1. C 12.140 f..755 4.CEl4 2.574 1.598 0.889
-2.0 11.7 96 6.411 3.739 2.730 1. 253 c. 54:3-3.( 11.:'57 5.872 3.70J 1.691 0.714 o.c-4.0 112.544 5.15<; 2.488 0.978 0.0 o.c-5.0 9.567 4.182 1.510 o.c o.c c.c-6.0 8.057 2.672 0.0 0.0 0.0 0.0
7 r' 5.385 C.C C.O C.G C.O 0.0- • v-.'~,
** 1J =20
123
(1=o.cO **
I.-No
'Wi-8.C -7.0 -6.0 -5 .. 0 -4.0 -3.0
4.0 1.022 1.022 1.072 1.022 1.01E 1. CO23.0 1.022 1. 02 2 1.022 1 .021 1.018 1.0012.Q 1. 017 1. C1 7 1. 017 I.C16 1.013 0.9961 (' 0.967 0.967 0.967 0.96t 0.963 0.946.v
0.0 0.751 0.751 0.751 0.751 0.747 C.73C-1.l C.377 C.377 C.377 0.377 0.373 0.354
-2.0 0.114 0.114 O.llL~ 0.114 C. 11 C C.C9C-3.C c. 024 0.024 0.024 0.023 0.020 C.C-4.C C.024 C.C04 ( • CCl, 0.C04 G.O 0.0-5.C 0.J01 O. CO 1 C.C01 C.C C.O C.C-6.0 (l.oon 0.000 0.0 o.e 0.0 o.c-7.e c.cco c.c c.o o.c O.C 0.0
** )) =20 ~ =0.80 **N
[..No
tN1. -8.0 -7.0 -6.0 -5.0 -4.0 - ~. 0
4.0 1. 045 1. C4: 1.04{' 1. C43 1.C35 1.0051.0 1.044 1. 04 4 1.04 l l 1.C42 1.C35 1.0 C52.0 l.O/tZ 1 .041 1.041 1.039 1.C32 I.C021.0 1.0C Q 1. CO 2 1. C08 I.C06 0.999 0.9690.0 0.836 0.835 C.835 C.P34 C.826 C.795
-1.0 0.475 0.475 0.475 0.473 0.46t C.433
-2.0 0.171) C.17C C.16S! 0.168 0.160 0.127-').0 0.043 0.043 C.043 0.C41 0.C34 c.c-4.0 0.010 0.009 0.0)9 0.C07 O.C c.o-5.0 0.OC2 C. CO2 C.GC2 c.c 0.0 0.0-6.0 0.000 o.ooc c.o c.c e.c c.c-7.0 O.r:JOO 0.0 0.0 0.0 0.0 c.c
** )} =20
124
~=1 .00 **i NO
LNi -R.O -7.0 -6.0 -5.0 -4.C -3.0
4.C 1.080 l.C79 1. 078 1.C74 1.060 1.0073.0 1.080 1.079 1.078 1.C74 1.C6C 1.0072.0 1.07R 1.078 1.077 1.073 1.C58 1.COt1.C 1.057 1.057 1.056 1.052 1.037 0.9840.0 C.923 0.923 0.922 C.91E 0.903 C.850
-1.0 0.5119 0.5A8 0.587 0.583 0.56E C.514
-2.0 0.247 0.247 C.246 0.242 0.227 0.172-3.G 0.076 0.07t C.075 C.C71 C.CSt C.C-4.0 C.020 00020 0.019 0.015 O.C O.G-5.C C.005 C.005 0.004 C.C O.C 0.0-6.0 0.001 00001 C.C C.C G.e C.C-7.0 0.000 0.0 0.0 0.0 o.c C.C
** ))=20
tfVr>
[/Vi. -8.C -7. C -6.e - 5.C -4.0 -3.0
4.0 1.135 1.134 1 .131 1.122 1.093 1.0083.e 10135 1.134 1. 1 31 1.122 1.093 1 .0082.0 1.135 1.133 1.130 1. 121 1.092 1. C07 _1.0 1.122 1 .120 1. 117 1.108 1.079 .0.S94O.C 1.022 1.021 1.0 If! 1.008 0.980 0.994
-1.0 0.722 0.721 C.718 C.708 0.t8C 0.593
-2.0 0.354 0.353 0.350 0.341 0.312 0.224-3.0 0.131 C.13C 0.127 0.117 0.088 G.O-4.0 0.043 0.042 00038 0.C29 c.c C.C-5.0 0.014 uo013 0.009 0.0 O.C C.Q-6.e ""1 ",,- C. (en 0.0 C.G o.~ 0.0v. '....., ..J:>-7.0 0.001 0.0 C.O c.c C.C C.C
**))=20
125
{,No
Ltv -8.C -7.e -6.0 -5.0 -4.0 -3.0i
4,,0 1.225 1.220 1.211 1.19C 1.136 1.e0530C 1.225 1.220 1.211 1 .. 19C 1.136 I.C05
20G 1.224 1.220 1.21]. 1. 19C 1.13(; I.C04100 1.216 1.212 1.203 1. 1R2 1.12E O.<;9£:
OoC 1.1 4 5 1.141 1.132 1. 11 C 1.057 C.925-1.0 C.AA6 c.eE2 C.873 0.852 0.79E 0.665
-20G 0.504 0.500 .0.491 0.470 0.416 0.282-3,,0 1J.2:?4 C.219 0.210 0.189 0.135 0.0-4,,0 0.089 0.DR4 0.075 c. C54 o.c c.o-500 0.035 0.030 0.021 0.0 o.c c.c-60C C.1113 0.009 0.0 0.0 0.0 0.0-7 00 C.004 C.c c.o C.C o.c C.C
** V =20 ~ =1.60 **N
1 DO
-5.0-600-7.C
-8.C
1.1731.1711.3731.3691.3191.104
0.7210.331C.1R30.0850.0390.014
-7.'0
1. 35 91.3591.3591.3541.305l.0ge
0.7070.367o. 16 e0.0710.02'4O.C
-6.0
1. 3 341.3141. 3341.33C1.2811.065
C.6A30.342C.1440.0470.0C.O
-5.0
1. 22 e1. 2881.2881.28"31.2341.019
O.t:3(;0.296G.C97o.co.c0.0
-4.0
1. 1911.1911. 1911. 18 (;1. 1370.922
0.53<10.1 99O.Go.cG.C0.0
-3.0
0.996C.99(;0.9950.991C.<;410.725
C.341O.GO.Co.cc.e0.0
**V =2C
126
~N=I.8C **
"N.0
'-N'1. -8.0 -7.0 -6.0 -5.0 -4.0 -3.0
4.0 1.632 1.581 1.524 1.425 1.257 C.(H83.0 1.63:? 1.581 1.524 1.425 1.257 G.cHE2.C 1.612 1.597 1.524 1.425 1.251 O.cHS1.0 1.610 1.584 1. 521 1.422 1.254 0.9750.0 1.597 1.552 1.488 1.389 1.222 0.943
-1.0 1.424 1.379 1.316 1.217 1.049 0.769
-2.0 1.055 1.01 C 0.941 0.1"48 C.tSC 0.399-3.0 0.65R 0.612 0.549 0.450 0.282 O.C-4.C e.176 G. 33 0 0.267 '0.168 0.0 o.e-5.0 C.208 0.162 C.099 c.c o.c c.e-6.0 0.108 0.063 0.0 o.c o.c c.c-7.0 S.045 O.G 0.0 0.0 0.0 0.0
**)} =20 0. =2.00 **N
"No
lwj. -R.f) -r.O -6.0 - 5.0 -4.0 -3.0
4.C ?112 1.973 1.816 1.613 1.333 0.9503.0 2.112 1.973 1.8H; 1. t 13 1.333 C.95C~.O 2.112 1.973 1.816 1.613 1.333 C.S5CI.e 2.110 1.972 1.R14 1...612 1.332 0.941:10.0 2.089 1. 951 1.793 1. 591 1.311 0.928
-La 1.956 1. au: 1.660 1.458 1. 177 C.794
-2.e 1.612 1.474 1. 316 1.114 0.834 0.449-3.0 1.165 1.027 0.969 C.t67 0.38t C.C-4.0 0.779 0.641 0.483 0.281 O.C C.C-5.0 8.49R 0.360 C.202 0.0 0.0 a .c-6.S 0.296 c. 1'5 C c.o c.c o.c c.c-7.0 0.1 38 G.O 0.0 o.c o.c c.c
127
APPENDIX B
PENALTY FACTORS FOR UNKNOWN DEMAND AND RESISTANCE PARAMETERS
. Ga/2Tables of the rat10 YAid S ~ G Ie for unknown demand and
, N, R' R
resistance parameters (see Equation V.32). The tables are for:
GSN=0.1,0.2,0.3
i d=-3,-6
v =5,10
~SN=O.6,1.0,1.4,1.8
iN =-8(1)-3a
128
**~N=O .. 10 -LcL=3.00 **** )} = 5 flf3
N=0.60 **
t No
iNi -8.0 -7.0 -6.0 -5.0 -4.0 -3.0
4.0 1.192 1. 175 1.156 1.132 1.098 1.0433.0 1.191 1.174 1.154 1.130 1.096 1.0422.0 1. 182 1 .. 165 1.146 1.122 1.088 1.034l.n 1.130 1 .. 113 1.094 1.070 1.035 0.981O. 0 0.914 0 .. €97 .C.877 0.853 0.818 0.761
-1.0 0.550 0 .. 533 0.513 .0.488 0.452 0.391
-2.0 0.291 0,,274 0.255 0.229 0.193 0.129-3.0 0'.1 E4 0 .. 147 0.127 0.102 0.065 0.0-4.t) 0.100 0.083 O.06? 0.037 0.0 0.0-5.0 0.062 0.045 0.025 0.0 0.0 0.0-6.0 0.037 0.020 0.0 0.0 0.0 0.0-7.0 0.017 0.0 0.0 0.0 0.0 0.0
** ()(3,... =0.10 -J.,cL=3.00 **
** V= 5 )-<(3, =1.00 **. 1\1
tNo
i,v.j.. -8.0 -7.0 -6.0 -5.0 -4.0 "':" 3. 0
q.O 2 .. 027 1.777 1.584 1.420 1.260 1.0843.0 2.027 1.776 1.584 1. 1120 1.260 1.0842.0 2.025 1.714 1.582 1.417 1.258 1.0811.0 2.003 1.752 1.559 1.395 1.236 1.0590.0 1.869 1 .. 618 1.425 1.261 1.101 0.923
-1.0 1.544 1. 293 1.100 0.935 0.774 0.593
-2.0 1.207 0.956 0.763 0.597 0.436 0.251-3.0 0.960 0.709 0.516 0.350 0.187 0.0-4.0 0.773 0.522 0.329 0.163 0.0 0.0-5.0 0.610 0.359 0.166 0.0 0.0 0.0-6.0 0.444 0.193 0.0 0.0 0.0 0.0-7.0 0.251 0.0 0.0 0.0 0.0 0.0
**(Jp., =0.10At
** )} = 5
129
-l-cL=3.00 **
)A-f3N
=1.40 **
"No
~N1 -8.0 -7.0 - 6.0 -5.0 -4.0 -3.0
4.0 7.946 4.772 3.142 2.2C8 1.599 1.1413.0 7.946 4.772 3. 142 2.208 1.599 1.1412.0 7.946 4.772 3.142 2.208 1.599 1.1401.0 7.938 4.763 3. 134 2.200 1.591 1.1320.0 7.866 4.692 3.062 2.128 1. 519 1.059
-1.0 7.615 4.441 2.811 1.877 1.267 0.805
-2.0 7.236 4.062 2.432 1.497 0.886 0.420- 3. 0 6.822 3.648 2.018 1.082 0.470 0.0-4.0 6.355 3.180 1. 550 0.614 0.0 0.0-5.0 5.742 2.567 0.936 0.0 0.0 0.0-6.0 4.806 1. 631 0.0 0.0 0.0 0.0-7.0 3.175 0.0 0.0 0.0 0.0 0.0
**crp.,,,,=0.10
** v= 5
- LcL=3.00 **
J'<I'N ::: 1.80 **
LNo
IN1. -8.0 -7.0 -6.0 -5.0 -4.0 -3.0
4.0 55.273 20.626 8.735 4.181 2.193 1.1873.0 55.273 20.626 A.735 4.181 2.193 1.1872.0 55.273 20.626 8.735 4.181 2.193 1.1871.0 55.270 20.623 8.732 4.178 2.190 1.1840.0 55.237 20.590 8.699 4.145 2.157 1.151
-1.0 55.071 20.423 8.532 3.978 1.990 0.981
-2.0 54.102 20.055 8.163 3.609 1.619 0.607-3.0 54.104 19.456 7.565 3.009 1.018 0.0-4.0 53.090 18.443 6.55 '/ 1.994 0.0 0.0-5.0 51.100 16.452 4.559 0.0 0.0 0.0-6.0 46.544 11. 895 0.0 0.0 0.0 0.0-1.0 34.651 0.0 0.0 0.0 0.0 0.0
** o-f3N
=0 B 10
** ))-=10
130
-Ld.=3.00 **
)A-(3N =0 • 6 0 **
'-No
&/111- - 8.0 -7.,0 -6.0 -5.0 -Q.O -3.0
4.0 1.088 1.087 1.085 1.081 1.071 1.0423.0 1.088 1. 087 1.085 1.080 1.070 1.0412.0 1.080 1.079 1.018 1.01~ 1.063 1.0341.0 1.027 1.026 1.024 1.019 1.009 0.9800.0 0.802 0.R01 .0 .. 799 0.795 0.184 0.753
-1.0 0.423 0.422 O.Q20 0.416 0.405 0.372
-2.0 O. 1St:} 0.158 0.156 0 .. 151 0.140 0.106-3.0 0.054 0.053 0.051 0.046 0.034 0.0-4.0 0.019 0.018 0.016 0.012 0.0 0.0-5.0 0.007 0.006 0.004 0.0 0.0 0.0-6.0 0.003 f).OO2 0.0 0.0 0.0 0.0-7.0 0 .. 001 0.0 0.0 0.0 0.0 0.0
**(J~1v=0.10
** V=10
-LcL=3.00 **
F/3 =1. 00 **. N
&N.0
LN1- - 8.0 -7.0 -6.0 -5.0 -4.0 -3.0
4 .. 0 1.258 1 .. 244 1.225 1., 197 1. 148 1.0553. f) 1.258 10 244 1 .. 225 1.196 1.14A 1 .. 0552.0 1.256 1 .. 24 '2 1.223 1.. 194 1 .. 146 1.0531.0 1.233 1. 219 1.2 CO 1.172 1.123 1.. 0300.0 ' .. f)94 1.080 '.061 1.. 032 0.984 0.890
-1.1) 0 .. 755 0.741 0.722 0.694 0 .. 645 0.549
-2.0 0.412 0.397 0.378 0.350 0.301 0.203-3.0 0.210 0 .. 196 0.177 0.148 0.09.9 0.0-4.0 0.112 0 .. 097 0.078 0.050 0.0 0.0-5.0 0 .. 062 0 .. 047 0.029 0.0 0 .. 0 0.0-6.0 0.033 0.019 0.0 0.0 0.0 0.0-7.0 0.014 0.0 0.0 0.0 0.0 0.0
131
**CJ1\,=O. 1 0 - lcL =3 • 00 **** v =1 0 F~ =1 • 40 **
N
l,NO
LNi. -8.0 -7.0 -6.0 -5.0 -4.0 - 3. 0
4.0 1.985 1.805 1.648 1.491 1.308 1.0713.0 1.985 1.805 1.648 1.491 1.308 1. 0712.0 1. 984 1. 804 1.648 1. 490 1.308 1.0701 .0 1.976 1.796 1.639 1.482 1.299 1.0620.0 1.901 1.722 1.565 1.407 1.225 0.987
-1.0 1.640 1.461 1.304 1. 1,46 0.963 0.724
-2.0 1.255 1.075 0.918 0.760 0.577 0.336-3.0 0.921 0.741 0.584 0.427 0.243 0.0-q.O 0.678 0.498 0.341 O.18q 0.0 0.0-5.0 O.lt94 0.315 0.158 0.0 0.. 0 0.0-6.0 0.337 0.157 0.0 0.0 0.0 0.0-7.0 0.180 0.0 0.0 0.0 0.0 0.0
**cr(3rv=C. 1 0 ~ LcL =3 • 0 0 **** )} =10 /(3/1/ == 1. 80 **
l,v;.0
LNi. -8.0 -7.0 -6.0 -5.0 -q.O - 3.0
4.0 5.970 4.0q2 2.918 2.165 1.580 1 .0693 .. 0 5.~nO 4.042 2.918 2.165 1.580 1.069200 ').970 4.042 2.918 2.164 1.580 1.0681 .0 5.968 4.039 2.915 2.162 1.578 1.066000 5.933 4.005 2.881 2.128 1.543 1.031
-100 5.759 3.831 2.707 1.954 1.369 0.856
-2 .. 0 5.386 3.458 2.33~ 1.580 0.995 0.480-3.0 4.909 2.981 1.857 1.103 0.518 0.0-4.0 4.392 2.464 1.34C' 0.586 0.0 0.0-500 3.806 1.878 0.754 0.0 0.0 0.0-6.0 3.052 1.124 0.0 0.0 0.0 0.0-7.0 1. 928 0.0 0.0 0.0 0.0 0.0
* *CJ{31v =0 .. 1 0
** )}= 5
132
- LJ=6.00 **
rf-lN
=0.60 **iNo
t N1. -8.0 -7.0 -6.0 -5.0 -4.0 -3.0
Q.O ,. 311 '.301 1.283 1.260 1.226 1.1113.0 1.315 1.300 1. 281 1.258 1.224 1.1692.0 1.305 1.289 1.211 1.248 1.214 1.1591.0 1.242 1.227 1.209 1.186 1.152 1.0960.0 0.992 0.916 ·0.958 0.934 0.900 0.841
-1.0 0.581 0.566 0 .. 547 ,0.523 0.487 0.425
-2.0 0.298 0 .. 283 0.264 0.240 0.203 0.138-3. a 0.163 00141 0.128 0.104 0 .. 061 0.0-4.0 0.096 0 .. 081 0.062 0.038 0.0 0.0-5.0 0.OS9 0.043 0.025 0.0 0.0 0.0- 6.0 0.034 0.019 0.0 0.0 0.0 0.0-7.0 0.016 0.0 0.0 0.0 0.0 0.0
** IT(3~ =a•1 0
** ).J= 5
-LcL=6.0n **
)A/3/V ='.00 **LNo
l'Ni -8.0 -7.0 -6.0 -5.C -4.0 -3.0
4.0 2. 110 1.881 1.700 1.540 1.381 1.2003.0 2.109 1.880 1. 699 1. 540 1.381 1. 2002. 0 2.101 1.878 1.697 1.537 1.378 1.1911.0 2.080 1.851 1.610 1.511 1.352 1.110o. (' 1.925 1.696 1.515 1.355 1.196 1.012
-1.0 1.559 1.330 1 .148 0.988 0.828 0.641
-2. n 1.1'10 0.961 0.779 0 .. 619 0.457 0.266- 3.0 0.921 0.698 0.517 0.356 0.193 0.0-4.0 0.735 0.506 0.324 O. 163 0.0 0.0-5.0 0.572 0.343 C.161 0.0 0.0 0.0-6.0 0.411 0.182 0.0 Q.. O 0.0 0.0-7.t) 0.229 0 .. 0 0.0 0.0 0.0 0.0
133
** CJl3rv==0.10 -l-cL=6.00 **** )j== 5 ~f-lN =1.40 **
'-'1\10
Ltv - 8.0 -7.0 -6.0 -S.O -4.0 -3.0:1
4.0 7.661 4.765 3.233 2.328 1.719 1. 2483.0 7.661 4.765 3.233 2.328 1.719 1.2482.0 7 • 660 4.764 3.232 2.327 1.719 1.2471.0 7.651 4.755 3.222 2.317 1.709 1.2370.0 7. ') 68 4.672 3.139 2.234 1.626 1.153
-1.0 7.286 4.390 2.857 1.952 1.342 0.867
-2.0 6.871 3.975 2.442 1.536 0.925 0.446-3.0 6.432 3.535 2.002 1.096 0.484 0.0-4.0 S.950 3.054 1.520 ').613 0.0 0.0-5.0 5.338 2.441 0.908 0.0 0.0 0.0-6.0 4.431 1.534 0.0 0.0 0.0 0.0-7.0 2.897 0.0 0.0 o.n 0.0 0.0
**~~N=0.10 -LJ=6.00 **** )) == 5 JA-j3N= 1.80 **
(...,0
LNj. -8.0 -7.0 -6.0 -5.C -4.0 -3.0
4.0 51.464 19.884 8.713 4.304 2.320 1.2853.0 51.464 19.884 8.713 4.304 2.320 1.2852.0 51.464 19.884 8.713 4.303 2.320 1.2851. I) 51.461 19.881 8.710 4.3CO 2.317 1.2820.0 51.423 19.843 8.672 4.262 2.278 1.243
-1.0 51.235 19.655 8.484 4.074 2.090 1.053
-2.0 50.832 19.252 A.081 3.671 1.685 0.644-3.0 50.198 18.618 7.446 3.035 1.048 0.0-4.0 49.155 17.575 6.403 1.991 0.0 0.0-5.0 47.168 15.587 4.41f~ 0.0 0.0 0.0-6.0 42.756 11. 175 0.0 0.0 0.0 0.0-7.0 31.584 0.0 0.0 0.0 0.0 0.0
134
**0-(3 =0.10 -L.l=6.00 **.IV C/..
** V=10 /A-pI\I=O.60 **L.IV.
0
L./Vi -8.0 -7.0 -6.0 -5:" 0 -4.0 -3.0
4.0 1 .217 1.217 1. 215 1.211 1.200 1.1713.0 1.217 1.216 1 .214 1. 21 0 1.199 1.1702.0 1.208 1.207 1.205 1.201 1. 191 1.1611. 0 1.144 1.143 1.141 1. 131 1. 126 1. 0910.0 0.883 0.882 .0.881 0.876 0.866 0.834
-1.0 0.456 0.455 0.453 0.449 0.438 0.405
-2.0 0.166 0.165 0.164 0.159 0.148 0.113-3.0 0.054 0.053 0.052 0.047 0.036 0.0-4.0 0 .. 019 0.018 0.016 0 .. 012 0.0 0.0-5. 0 0.007 0.006 0.004 0.0 0.0 0.0-6.0 0.003 0.002 0.0 0.0 0.0 0.0-1.0 0.001 0.0 0.0 0.0 0.0 0.0
**()~1\I=0.10 -LcL=6.00 **
** )} =1 0 ~(31./ =1 • 0 0 **[NO
L.}"I - 8.0 -7.0 -6.0 -5.0 -4.0 -3.0:1.
4.0 1.374 1.360 1.343 1.315 1.266 1.1713.0 1.373 1.360 1.342 1.315 1.266 1.1112.0 1.371 1.35A 1.340 1.. 312 1 .. 264 1.1681. 0 1 .. 344 1.330 1.313 1.285 1.236 1. 1410.0 1.. 182 1. 169 1.152 10 124 1.075 0.979
-1.0 0.801 0.188 0.770 0 .. 743 0.693 0.595
-2.0 0 .. 425 0.412 0.394 0.366 0.317 0.216-3.0 0.210 0.197 0.179 0.151 0.102 0.0-4.0 0.109 0.095 0.078 0.050 0.. 0 0.0-5.0 0.059 0 .. 046 0.028 0 .. 0 0.0 0.0-6.0 0.031 0.018 0.0 0.0 0.0 0.0-7.0 0.013 0.0 0.0 0.0 0.0 0.0
135
**CT(3/v =0.10 -(,cL =6.00 **
** )}=10 )A-j3/V =1.40 **
l-No
l-N:1. -8.0 -7.0 -6.0 -5.0 -4.0 - 3.0
4.0 2.066 1.902 1.754 1.602 1.419 1.1753.0 2.066 1.902 1. 754 1.602 1.419 1.1752.0 2.066 1.902 1.754 1.601 1.419 1.1741 • 0 2.056 1.892 1.744 1.591 1.409 1.1640.0 1.970 1.806 1.658 1.505 1.323 1.078
-1.0 1.676 1.512 1.364 1.211 1.028 0.782
-2.0 1.254 1.090 0.942 0.789 0.606 0.357-3.0 0.899 0.735 0.587 0.434 0.251 0.0-4.0 0.649 0.485 0.337 0.184 0.0 0.0-5.0 0.465 O. 301 0.153 0.0 0.0 0.0-6.0 0.112 0.148 0.0 0.0 0.0 0.0-7.0 o. H4 0.0 0.0 0.0 0.0 0.0
**CT(3/v =0.10 -CoL =6.00 **
** )) =1 C JA-f3N :: 1 • 8 0 **LNO -
L-Nj.. - 8.0 -7.0 -6.0 -5.0 -4.0 - 3.0
4.0 5.819 4.060 3.00:2 2.272 1.688 1.1613.0 5.819 4.060 3.00:2 2.272 1.688 1. 1612.0 5.819 4.059 3.002 2.271 1.688 1.1611.0 5.816 4.056 2.999 2.268 1.684 1. 1570.0 5.776 4.017 2.960 2.229 1.645 1.118
-1.0 5.581 3.821 2.764 2.033 1.449 0.921
-2.0 5. 173 3.413 2.356 1.625 1.040 0.510-3.0 4.666 2.907 1.849 1.118 0.533 0.0- 4.0 4. 134 2.374 1.317 0.586 0.0 0.0-5.0 3.548 1.789 0.731 0.0 0.0 0.0-6.0 2.817 1.057 0.0 0.0 0.0 0.0-7~0 1.760 0.0 0.0 0.0 0.0 0.0
136
**cr(3/V = o. 2 0 - ~d = 3 • 0 0 **** J)= 5 /'-j3N =0.60 **
lNO
LJ\/. -8.0 -1.0 -6.0 -5.0 -4.0 - 3.0::1.
4.0 1.318 1.295 1.271 1.245 1.210 1.1513.0 1.315 1.292 1.269 1.242 1.207 1.1542.0 1.302 ' 1.279 1.256 1.230 1.195 1.1411.0 1.233 1.210 1.187 1. 160 1. 125 1.0100.0 0.975 0.952 . 0.928 0.901 0.865 0.809
-1.0 0.51lJ 0.551 0.528 0.500 0.463 0.402
-2.0 0.306 0.283 0.259 0.232 0.194 0.130-3.0 0.178 00 155 0.131 0.103 0.065 0.0-4.0 0.114 0.090 0.067 0.039 0.0 0.0-5.0 0.075 0.052 0.028 0.0 0.0 0.0-6.0 0.047 0.024 0.0 0.0 0.0 0.0-7.0 0.023 0.0 0.0 0.0 0.0 0.0
**O""f->N =0.20 - Lc/..=3. 00 **
* ,. }) = 5 /lj3,.y =1 • 0 0 * ,.
ltV0
IN:L -8.0 -1.0 -6.0 -5.0 -4.0 -3.0
4.0 2.261 1.91q 1.687 1.506 1.341 1.1643.0 2.260 1.919 1.686 1.505 1.341 1.1642.0 2.257 1. 915 1. 683 1.502 1.337 1.1601.0 2.227 1.886 1.653 1.412 1.307 1.1300.0 2.068 1.126 1.494 1.313 1.147 0.969
-1.0 1.711 1.369 1.137 0.955 0.789 0.607
-2.0 1. 362 1.020 0.788 0.6e5 0.438 0.252-3.0 1.114 0.772 0.539 0.357 O. 188 0.0-4. 0 0.926 0.584 0.352 0.169 0.0 0.0-5.0 0.158 0.416 C.183 0.0 0.0 0.0-6.0 0.515 0.233 0.0 0.0 0.0 0.0-1.0 0.342 0.0 0.0 0.0 0.0 0.0
137
** CJf3",=0.20 -L-d=3.00 **
**)}= 5 /Af3N
==1.lJO **
i lVO
LN -8.0 -1.0 -6.0 -5.0 -4.0 -3.01-
4.0 9.636 5.292 3.320 2.289 1.651 1.1973.0 9.636 5.292 3.319 2.289 1.651 1.1972.0 q .635 5.291 3.319 2.288 1.656 1.1961.0 9.625 5. 281 3.308 2.277 1.646 1.1850.0 9.540 5.196 3.223 2.192 1.560 1.099
-1.0 9.265 4.921 2.948 1.917 1.285 0.820
-2.0 A.A73 4.529 2.555 1.524 0.890 0.422- 3.0 8.457 4.113 2.139 1.108 0.472 0.0-4.0 7.987 3.642 1.669 0.637 0.0 0.0-5.0 7.351 3.007 1.033 0.0 0.0 0.0
. -6.0 6. 319 1.974 0.0 0.0 0.0 0.0-7.0 !.J.345 0.0 0.0 0.0 0.0 0.0
**a-~1\I=0.20 -LcL==3.00 **
** )} = 5 )A~ =1.80 **IV
LIvO
LAId.- -8.0 -7.0 -6.0 -S.C -4.0 -3.0
4. 0 71.400 23.719 9.338 4.301 2.237 1.2253.0 11.400 23.178 9.338 4.301 2.236 1.2252.0 71.399 23.778 9.33R 4.301 2.236 1.2241.0 11.396 23.775 9.335 4.2S7 2.233 1.2210.0 71.357 23.136 9.296 4.258 2.194 1.182
-1.0 71.175 23.554 9.11 Ll LJ.076 2.011 0.997
-2.0 70.794 23.173 8.732 3.694 1.628 0.610-3.0 70. 193 22.512 8.13 'j 3.092 1.024 0.0-4.0 69.17!.J 21.552 7. 111 2.071 0.0 0.0-5.0 67.106 19.484 5.042 0.0 0.0 0.0-6.0 62.068 14.445 0.0 0.0 0.0 0.0-7.0 47.626 0.0 0.0 0.0 0.0 0.0
**0-(3N=0.20
** )}=10
138
-LcL=3.00 **JA-f3
N=0.60 **
LN0
LNi - 8.0 -7.0 -6.0 -5.0 -4.0 -3.0
4.0 1.204 1.203 1.200 1.196 1.185 1.1573.0 1.203 1. 201 1.199 1 .. 194 1.184 1.1552.0 1.192 1.190 1. 188 1. 184 1.173 1.1441.0 1.120 1. 119 , .116 1. 112 1.101 1.0720 .. 0 0.8")2 0 .. 851 -0.8tla O.8tl4 0.833 0.802
-1 .0 0.436 0.434 0.432 0.427 0.416 0.383
-2.0 O. 161 0.160 0.157 0.153 0.141 0.107-3.0 0.055 0.054 0.051 0.047 0.035 0.0-4.0 0.021 0.019 0.017 0.012 0.0 0.0-5.0 0.009 0.007 0 .. 005 0.0 0.0 0.0-6.0 0.004 0.002 0.0 0.0 0.0 0.0-7.0 0.001 0.0 0.0 0.0 0.0 0.0
** )}=10 ./'-'/3. = 1.00 **.1Jr
LN-0
LNi. - 8. 0 -7.0 -6.0 -5.C -4.0 -3.0
4.0 1.354 1.334 1.311 1.280 1. 230 1.1373.0 1.353 1.334 1.311 1.280 1.230 1.1372.0 1.350 1.331 1. 308 1.277 1.227 1.1341.0 1.320 1. 300 1.278 1.247 1.196 1.1030.0 1. 155 1. 135 1. 112 1.081 1.031 0.937
-1.0 0.783 0.764 0.741 0.710 0.659 0.563.-2.0 0.427 0.407 0.384 0.353 0.302 0.204-3.0 0.224 0.204 0.182 0.150 0.099 0.0-4.0 O. 125 0.105 0.083 O. 051 0.0 0.0-5.0 0.074 0.054 0.031 0.0 0.0 0.0-6.0 0.042 0.023 0.0 0.0 0.0 0.0-7.0 0.020 0.0 0.0 0.0 0.0 0.0
139
** G""{:l,v= 0.20 - ~cL = 3.00 **
** )} =10 ~f3N =1.40 **LNo
L.v~ -8.0 -7.0 -6.0 -5.0 -4.0 -3.0
n.o 2.162 1.917 1.728 1.555 1.366 1.1283.0 2.162 1.917 1.728 1.555 1.366 1.1282.0 2.161 1. 91 6 1.727 1.5~4 1.365 1.1271.0 2.150 1.905 1.716 1.~43 1.354 1. 1160.0 2.062 1. 817 1.628 1.455 1.266 1.02A
-1.0 1. 776 1.531 1.342 1.1,69 0.980 0.740
-2.0 1.377 1. 132 0.943 0.770 0.580 0.338-3.0 1.041 0.796 0.607 0.434 0.244 0.0-4.0 0.797 0.553 0.364 0.190 0.0 0.0-5.0 0.607 0.362 0.173 0.0 0.0 0.0-6. " 0.433 0.189 0.0 0.0 0.0 0.0-7.0 0.244 0.0 0.0 0.0 0.0 0.0
**CT(3,v=O.20 -LcL =3.00 **
** V =10 ~j3 =1.80 **N,
t No
LNi -8.C -7.0 -6.0 -5.0 -4.0 -3.0
4.0 7.051 4.416 3.057 2.227 1.621 1.1073.0 7.051 4.416 3.057 2.227 1.621 1. 1072 .. 0 7.051 4.416 3.0Ci7 2.226 1.621 1 .. 1071.0 7.048 4.412 3.05,~ 2.223 1.617 1.103
•0.0 7.007 4.372 3.013 2.183 1.577 1.063-1 .. 0 6.817 4. 182 2.823 1.. 993 1. 387 0.871
-2 .. 0 6.431 3.796 2.437 1.606 1.000 0.483-3.0 5.952 3.316 1.958 1.127 0.520 0.0-4.0 5.432 2.797 1.438 0.607 0.0 0.0-5.0 4.825 2. 190 0.83" 0.0 o.c 0.0-6 .. 0 3.994 1.359 0.0 0.0 0.0 0.0-7.0 2.636 0.0 0.0 0.0 0.0 0.0
**G"J3tv -= 0 • 2 0
** )} = 5
140
-Lcl.. =6.00 **
~f3N=0.60 **
/-N'o
L-N-j. -8.0 -7 .. 0 - 6.0 -5.0 -4 .. 0 -3.0
4. 0 2.013 1.997 1.979 1.957 1.924 1.8693.0 2.001 1.991 1.973 1. 950 1.911 1.8622.0 1.978 1.962 1.944 1.921 1.888 1.8331..0 1.831 1.821 1.803 1.780 1.741 1 .. 6900.0 1.311 1.. 355 -1.337 1.313 1.278 1..217
-1.0 0.123 0.707 0.689 0.fi65 /).628 0.560
-2.0 0.336 C.320 0.301 0.277 0.239 0.167- 3.0 0.172 0.156 e.137 0.112 0.014 0.0-4.0 0.099 0.082 0.064 0.039 0.0 0.0-5.0 0.060 0.044 0.025 0.0 0.0 0.0-6.0 0.035 0.019 0.0 0.0 0.0 0.0-7.0 0 .. 016 0.0 0.0 0.0 0.0 0.0
**CTp.,J\I=0.20
** )} -= 5
- icL=6.00 *:lr
/Ap"", -= 1 • 0 0 **LNO
L,y1. -8.0 -7.0 -6.0 -5.0 -4.0 -3.0
4.0 2.712 2.475 2,,293 2. 134 1.971 1.7763.0 2.711 2.414 2.292 2.133 1.970 1.7752.0 2.703 2.466 2.284 2.125 1.961 1.7661.0 2.644 2.IW7 2.225 2.0 E5 1.902 1.7060.0 2.357 2.120 1.938 1 .. 778 1.614 1.415
-1.0 1.782 1. 545 1.363 1.202 1.036 0.832
-2.0 1. 280 1.042 0.860 0.699 0.531 0.322-3.0 0.963 0.725 0.543 0.381 0.213 0.0-4.0 0.751 0.513 0.331 0.169 0.0 0.0-5.0 0.582 0.345 0.162 0.0 0.0 0.0-6.0 0.421 o. 183 0.0 0.0 0.0 0.0-7.0 0.238 0.0 0.0 0.0 0.0 0.0
** CJ"f3N
=0.20
** )} = 5
141
- L./=6.00 **c_
)A-f3 =1.40 **IV
l'No
IN -8.0 -7.0 -6.0 -5.C -4.0 -3.01-
4.0 A.332 5.325 3.784 2.875 2.248 1.733300 8.332 5.325 3.783 2.875 2.247 1.7332.0 8.330 5.323 3.781 2.873 2.245 1.7311 0 0 8. 308 5.301 3.760 2.852 2.224 1.7090.0 8.156 5.149 3.608 2.699 2.071 1.554
-100 7.715 4.708 3.167 2.2.58 1.628 1.107
-2.0 7.153 4.146 2.604 1.694 1.063 0.536-300 6.624 1.617 2.075 1.165 0.532 0.0-4.0 6.095 3.088 1.5l.6 0.635 0.0 0.0-5.0 S.462 2.454 0.9'12 0.0 0.0 0.0-6.0 4.551 1.543 0.0 0.0 0.0 0.0-7 0 0 3.008 0.0 0.0 0.0 0.0 0.0
** V = 5
-L.c/..=6.00 **
JA-p..y = 1.80 **
L-NO
{..yd. -8.0 -7.0 -6.0 -5.0 -4.0 -3.0
4. a 53.384 20.544 9.3 C3 4.877 2.829 1.6983.0 53.384 20.544 9.301 4.877 2.82q 1.6982.0 53.383 20.544 9.3 C3 4.876 2.829 1.697'.0 53.377 20. 537 9.296 4.870 2.822 1 .691O.C' 53.307 20.467 9.227 4.800 2.752 1.620
- 1. a 53.016 20.176 8.935 4.508 2.459 1. 324
-2.0 52.472 19.632 8.391 3.963 1.913 0.772-3.0 51.711 18.871 7.6310 , 3.201 1.148 0.0-4.0 50.568 17.728 6.486 2.056 0.0 0.0-5.0 48.516 15.675 4.432 0.0 0.0 0.0-6.0 44.087 11. 245 0.0 0.0 0.0 0.0-7.0· 32.844 0.0 0.0 0.0 0.0 0.0
142
**CT{3N=O .. 20 -L-d.=6.00 **
** )J=10 / f3-v =0.60 **
~1Vo
t/Vi. -8.0 -7 .. 0 -6.0 -5.0 -4.0 - 3.0
4.0 1.922 1.922 1.920 1.916 1.906 1 .. 8763.0 1. 919 1.918 1.916 1.912 1.902 1.8732.0 1.894 1.893 1.892 1.888 '.871 1.8481. 0 1 e7 49 1 .. 748 1.747 1 .. 743 1 .. 732 1.1020.0 1.265 1. 264 ·10263 1e259 L 248 1.215
-1eO 0.592 Oe591 0.589 0.585 0.513 0.531
-2. a 0.195 0 .. 194 0.192 0.188 0.176 0.137-3.0 0.059 0.058 Oe056 0.052 00039 0.0-4.0 0.019 0.018 Oe016 Oe012 0.0 0.0-5.0 0.001 0.006 0.004 0.0 0.0 0.0-6.0 0 .. 003 Oe002 0.0 OeO 0.0 0.0-7.0 0.001 0.0 0.0 0.0 0.0 0.0
** CJf3N=0.20 - Ld..=6. 00 ***'" v = 10 /A;3/V = 1. 0 n *'"
l,.y,0
LIV:! -8.0 -7 .. 0 -6. C -5.0 -4.0 -3eO
4.0 1e 961 1.948 1.930 1.902 1.852 1.7503.0 1 .. 961 1. 941 1.929 1.902 1.852 1.1492.0 1.954 1.940 1.922 1.895 1.845 1.1421.0 1.892 1.879 1.861 1.833 1.184 1.6800.0 1.595 1.581 1.563 1.536 1.485 1..381
-1.0 0.996 0.983 0.965 Oe937 0.886 0.778
-2.0 0.483 0.470 o e452 0.424 0.373 0.262- 3.0 0.223 0.210 0 .. 192 0 .. 164 0.112 0 .. 0-4.0 0.111 0.097 0.080 0.052 0.0 0.0-5.0 0.059 0.046 0.028 0.0 0 .. 0 0 .. 0-6.0 0.032 0.018 C.o 0.0 0.0 0.0-1.0 0.014 0.0 0.0 0.0 0.0 0.0
* *CT[31\1 -=0. 20
** v=1 C
- LcL -= 6. 00 **
{No
[Ni - 8.0 -7.0 -6.0 -5.0 -4.0 -3.0
4.0 2.581 2.411 2.262 2~109 1.920 1.6533.0 2.~81 2.411 2.262 2.1C9 1.920 1.6S32.0 2.579 2.409 2.260 2.107 1.918 1.6511.0 2.557 2.387 2.238 2.085 1.896 1.6280.0 2.399 2.229 2.080 1.927 1.738 1.469
-1.0 1.940 1.770 1.621 1.468 1.278 1.007
-2.0 1.367 1. 197 1.049 0.895 0.705 0.431-3.0 0.939 0.769 0.620 0.467 0.276 0.0-4.0 0.663 0.493 0.345 O. 191 0.0 0.0-5.0 0.473 0.302 0.154 0.0 0.0 0.0-6.0 0.319 0.149 0.0 0.0 0.0 0.0-7.0 0.170 0.0 0.0 0.0 0.0 0.0
**U""(1,y =0. 20
** ))-=10
-tel =6.00 **
.1-';91\/=1.80 **lN
O
LN :! - 8.0 -7.0 -6.0 -5.0 -4.0 -3.0
4.0 n.360 4.534 3.470 2.736 2. 132 1.5543.0 6.360 4.534 3.4·J 0 2.736 2.132 1.5542.0 6.360 4.533 3.470 2.736 2.132 1.5541.0 6.353 4.~26 3.463 2.729 2.125 1.5470.0 f).280 4.454 3.391 2.657 2.053 1.474
-1. a 5.977 4.1'10 3.087 2.353 1.749 1.169
-2.0 5.425 3.599 2.516 1.801 1.196 0.613-3.0 4.816 2.989 1.926 1.192 0.586 0.0-4 .. 0 4.230 2.40lt 1.341 0.6e6 0.0 0.0-5.0 3.62lt 1.798 0.735 0.0 0.0 0.0-6.0 2.890 1.C63 0.0 0.0 0.0 0.0-7.0 1. R26 0.0 c.o 0.0 0.0 0.0
144
**0-(3AI =0 .. 30 - [d =3 • 00 **** )} = 5 JAf3
1V=0.60 **
LiV-a
i IV1 -B.O ':"'7.0 -6.0 -5.0 -q.o -3.0
4.0 1.5B6 1.548 1.516 1.485 1.449 1 .3913.0 1.580 1.541 1.510 L479 1.442 1.3912.0 1.553 1.515 1.483 1. 452 1.416 1.3641..0 1.441 1.402 1.311 1.340 1.303 1.2500.0 1.. 09/.J 1 .055 . 1.024 0.992 0.954 0.898
-1.0 0.625 0.E)86 0.5~q 0.522 0.482 0.421
-2.0 0.339 0.300 0.268 0.236 0.195 O. 131-3.0 0.210 0.171 O. 139 0.106 0.065 0.0-4.0 0.145 0 .. 106 0.074 0.041 0.0 0.0-5.0 0.104 0.065 0.033 0.0 O.c 0.0-6.0 0.071 0.032 0.0 0.0 0.0 0.0-7.0 0.039 0.0 0.0 0.0 0.0 0.0
**CT(3,v=0.30 -L-cL=3.00 **** V = 5 )Afi/V =1.00 **
LA/a
[,vi -8.0 -7.0 -6.0 -5.0 -4.0 - 3. 0
4.0 2.783 2. 209 1.891 1.678 1.503 1 .3211.0 2.782 2.207 1.889 1.677 1.502 1.3252.0 2.774 2.200 1.882 1.669 1.495 1.3181.0 2.727 2.153 1.835 L622 1 .. 441 1.2700.0 2.514 1.940 1.621 1.408 1.233 1.054
-1.0 2.098 1.523 1.205 0.9~1 0.814 0.631
-2.0 1.721 1.152 0.833 0.619 0.442 0.254-3.0 1.476 0.91)1 0.583 0.368 0.190 0.0-4.0 1.2R7 0.712 0.394 0.179 0.0 0.0-5.0 1.108 0.533 0.215 0.0 0 .. 0 0.0-6.0 0.894 0.319 0.0 0.0 0.0 0.0-1.0 0.575 0.0 0.0 0.0 0.0 0.0
145
**o-l3rv=0.30 -l-cL=3.00 **
** J) = 5 )-<~A/ =1.40 **
£-4'0
LN-i -8.0 "-7.0 -6.0 -5.0 -4.0 -3.0
4.0 13.726 6.373 3.656 2.4ft1 1.769 1.3063.0 13.726 6.373 3.6'56 2.440 1.769 1.3052.0 13.724 6.371 3 .6'54 2.439 1.767 1.3041.0 13.707 6.354 3.637 2.422 '.750 1.2860.0 13.594 6.241 3.524 2 .. 3.08 1.637 1.172
-1.0 13.275 5.922 3.205 1.989 1.316 0.848
-2.0 12.859 5.506 2.788 1.572 0.898 0.425-3.0 12.439 5.087 2.369 1.152 0.477 0.0-4.0 11. 965 4.612 1.894 0.677 0.0 0.0-5.0 11.290 3.9~6 1.218 0.0 0.0 0.0-6.0 10.072 2.719 0.0 0.0 0.0 0.0-7.0 7.354 0.0 0.0 0.0 0.0 0.0
**CJ[3ty =0 • 30 - l cr= 1 • 00 **** v= 5 /Aj3,1/=1.80 **
[NO
LIV1- - 8.0 -7.0 -6.0 -5.0 -4.0 -3.0
4.0 111.653 30.482 1 C. 4 82 4.517 2.317 1.2963.0 111.653 30.482 10.482 4.517 2.317 1.2962.0 111.652 30.u81 10.482 4.516 2.316 1.2951.0 111. 647 30.476 10.476 4.511 2.311 1.2900.0 111.595 30.424 10.425 4.459 2.259 1.237
-1.0 111.384 30.213 10.214 4.248 2.047 1.023
-2.0 110.981 29.810 9.810 3.844 1.642 0.614-3.0 110.376 2<3.205 9.20S 3.238 1.034 0.0-4.0 109.347 28.176 8.175 2.207 0.0 0.0-5.0 107.144 25.C372 5.970 0.0 0.0 0.0-6.0 101. 177 20.004 0.0 0.0 0.0 0.0-7 .. 0 81.177 0.0 0.0 0.0 '0. Q 0.0
146
**CJI3I\t=0.30 -LcL=3.00 **
** )} =10 JAf3N
=0.60 **
LNo
[IV:!.. - 8.0 -7.0 -6.0 -5.0 -4.0 -3.0
4.0 1. 448 1.446 1.443 1.438 1.426 1.3993.0 1.411: 10442 1.439 1.434 1.423 1.3952.0 1.422 1.420 1.417 1.412 1.401 1.3731.0 1.307 1.304 1 .301 1.296 1.285 1.2570.0 0.946 0.944 -0.941 0.936 0.924 0.894
-1.0 0.458 0.456 0.453 0.447 0.435 0.402
-2.0 0.166 0.163 0.16') 0.155 o. '42 0.108-3.0 0.059 0.056 0.053 0.047 0.035 0.0-4.0 0.024 0.021 0.018 0.013 0.0 0.0-5.0 0.011 0.009 0.006 0.0 0.0 0,.0-6.0 0.005 O.OC3 0.0 0.0 0.0 0.0-7.0 .0.002 0 .. 0 0.0 0.0 0.0 0.0
**CJ~N=O.30-Ld=3.00 **
** )} = 10 /A-f3/V =1.00 **
lNO
l,vi. -8.0 -7.0 -6.0 -Ci.O -4.0 -3.0
4.0 1. 548 1. 516 1.485 1.449 1.396 1.3033.0 1.547 1. 515 1.484 1.448 1.395 1.3022.0 1.541 1.50Q 1.478 1.442 1.389 1.2961. 0 1.492 1.460 1.429 1.393 1.340 1.2470.0 1.271 1.239 1.208 1.172 1.118 1.024
-1.0 0.83T 0.e05 0.774 0.738 0.684 0.588
-2.0 0.459 0.426 0.395 0.359 0.305 0.206-3.0 0.254 0.221 0.190 0.154 0.100 0.0-4.0 0.154 o. 122 0.091 0.054 0.0 0.0-5.0 0.100 0.067 0.037 0.0 0.0 0.0-6.0 0.063 0.031 0.0 0.0 0.0 0.0-7.0 0.033 0.0 0.0 0.0 0.0 0.0
147
**U(3N=0.30 -Ld=3.00 **
** )J =10 ,,/Af3,v ='u.40 **
llVo
[.l1/i -8.0 -7.0 -6.0 -5.0 -4.0 -3.0
4.0 2.S50 2.140 1.882 1.679 1.479 1.2393.0 2.549 2.140 1 .881 1.678 1.478 1.2392.0 2.548 2. 138 1.880 1.677 1.471 1.2381.0 2.S30 2.120 1.862 1.659 1.459 1.2200.0 2.413 2.003 1.745 1.542 1.342 1.102
-LO 2.080 1.671 1.412 1.209 1.009 0.167
-2.0 1.657 1.247 0.989 0.786 0.S8S 0.341-3.0 1.319 0.909 0.650 0.441 O. 246 0.0-4.0 1.073 0.663 C.4 C5 0.201 0.0 0.0-5.0 0.871 0.462 0.203 0.0 0.0 0.0-6.0 0.668 0.258 0.0 0.0 0.0 0.0-7.0 0.410 0.0 0.0 0.0 0.0 0.0
**Uj3I\r=0.30 -Ld. =3.00 **
** V =1 0 JAJ31\r =1. 80 **
lJ\/o
L,v:1 -8.0 -7.0 -6.0 -5.0 -q.O -3.0
4.0 9.631 5.185 3.318 2.340 1.697 1. 1193.0 9.631 5.185 3.318 2.340 1.n97 1 .1792.0 9.636 5.185 3.318 2.3110 1.697 1.1191. 0 9.631 5.179 3.312 2.334 1. 691 1.1730.0 9.571 5.126 3.259 2.281 1.638 1.119
-1.0 9.357 4.906 3.039 2.061 1.418 0.898
-2.0 8.9119 4.q98 2.630 1.652 1.009 0.487-3.0 8.466 4.014 2.147 1.169 0.525 0.0-4.0 7.942 3.490 1.623 0.645 0.0 0.0-5.0 7.291 2.846 0.918 0.0 0.0 0.0- 6.0 6.319 1.867 0.0 0.0 0.0 0.0-7.0 4.451 0.0 0.0 0.0 0.0 0.0
148
**CTf3N=Oe30 -L,d=6eOO **** )} = 5 JAj3,y =Oe 60 **
l~0
tAli -8.0 -7.0 -6.0 -5.0 -4.0 - 3e 0
4eO 4.559 4.543 4.526 4.506 4.479 4.4363.0 4.502 4.486 4.469 4.449 4.421 4.3782e 0 4.332 4. 316 4.299 4.279 4.251 4.205100 3.779 3.763 3.746 3.725 3.695 3.6440.0 2.462 2.445 ·2.428 2.405 2.371 2.307
-1.0 1.070 1.051 1.035 1.011 0.972 0.894
-2.0 0.415 0.3<38 0.380 0.355 0.315 0.230-3.0 0.189 0 .. 172 0.153 0.128 f).087 0.0-4.0 0.102 0.085 0.066 0.041 0.0 0.0-5.0 0.061 0.04/.t 0.025 0.0 0.0 0.0-6.0 0.036 0 .. 019 0.0 0.0 0.0 0.0-7.0 0.017 0.0 0.0 0.0 0.0 0.0
**()~N=O.• 30 -LcL=6.00 **
** ))= S .J-if/JV =1.00 **
l-NO
tNi -8.0 -7.0 -6.0 -5. a -4 eO -3.0
4 eO 4.622 4.310 4.188 4.029 3e862 3.6473.0 4e 611 4e360 4 e177 4.018 3e8S1 3.6372eO 4.564 4.312 4.130 3.971 3.803 3 e5881 e O 4.334 4.082 3.900 3e740 3e572 3e355OeO 3,,531 3.279 3.096 2e 9 36 2.765 2.540
-1.0 2.306 2.054 1.871 1.7C9 1.535 1.297
-2.0 1e 463 1. 211 1.027 0,,865 Oe687 0.442-3.0 1 .. 02 8 0.776 Oe592 0.429 Oe 250 0.0-4.0 0,,779 0.527 0.3Q2 0.179 0.0 0.0-5.0 0.600 0.348 0.163 0.0 OeO 0.0-6.0 Oe437 0.184 OeD 0.0 0.0 0.0-7.0 0.253 0.0 0.0 0.0 0.0 0.0
149
**a-{3N =0.30 -l.-cl.. =6.00 **
** ~ = 5 ~PN ==1.40 **
lJl1,
Ltv~ -R.O -7.0 -6.0 -5.0 -4.0 -3.0
4.0 10.076 6.873 5.318 4.404 3.744 3.1523.0 10.074 6.872 5.316 4.403 3.743 3.1512~ (I 10.063 6.860 5.305 4.391 3.731 3.1391.0 9.9RO 6.778 5.222 Q.309 3.648 3.055O. 0 9.558 6.355 -4.800 3.886 3.22q 2.627
-1.0 8.628 5.425 3.8 E9 2.954 2.289 1.683
-2.0 7.691 4.488 2.931 2.015 1.348 0.732-3.0 6.970 3.767 2.210 1.293 0.623 0.0-4.0 6.350 3.147 1.589 0.672 0.0 0.0-5.0 5.679 2.476 C.918 0.0 0.0 0.0-6.,0 4.762 1.558 0.0 0.0 0.0 0.0-7 Q O 3.204 0.0 0.0 0.0 0.0 0.0
** CTF->N=0.30 -L.cL =6.00 **
** )} = £) /Aft"", =1.80 **
LNo
LN:i. -8.0 -7.0 -6.0 -5.0 -4.0 - 3. a
4.,0 57.150 22.086 10.721 6.272 4. 114 2.8023.,0 57.150 22.085 10.727 6.272 4.113 2.8022.0 57. 148 22.C83 10.724 6.27C 4.111 2.7991.,0 57.122 22.057 10.699 6.2LJ4 4.085 2.7730.0 56.931 21.866 10.5e7 6.052 3.893 2.579
-1.0 56.321 21. 256 9.897 5.442 3.280 1.960
- 2. (I 55.422 20.357 8.997 4.541 2.376 1.047-3.0 54.390 19.324 7.964 3.507 1. 339 0.0-4.0 53.057 17.991 6.631 2.171 0.0 0.0-5.0 50.889 1c).823 4.462 0.0 0.0 0.0-6.0 46.431 11.364 0.0 0.0 0.0 0.0-7.0 35.069 D.C 0.0 0.0 0.0 0.0
150
**crf3/V=O~3C -Lc1=fi.OO **
** )) =10 .JAP.v =0.60 **
l/Vo
£,,vi -8 .. 0 -7 .. 0 -6.0 -5.0 -4.0 -3.0
4.0 4.485 4.484 4.483 4.479 4.471 4.4483.0 4~454 4.453 4.451 4.448 4.439 4.4162.0 4.312 4.311 4.309 4.3C6 4.297 4.2741.0 3.744 3.743 3.741 3.738 3.729 3.7020.0 2.375 2.374 .2.372 2 .. 368 2.358 2.323
-1 .. 0 0 .. 928 O~ 927 0.925 Q.. 921 0.909 0.861
-2.0 0.256 0 .. 255 0.253 0.249 0.236 0.190-3.0 0.067 0 .. 066 0.064 0.060 0.047 0.0-4.0 0.020 0.019 0.017 0.013 0.0 0.0-5 .. 0 0,,007 0.006 0.004 0.0 0.0 0.0-6.0 0.003 0.002 0.0 0.0 0 .. 0 0.0-7.0 0.001 0.0 0.0 0.0 0.0 0.0
**0(3/V =0. 3C - Ld=6.00 **
** V =10 "pf3/V =1.00 **
tlVo
L,v:1. -8.0 -7.0 -6.0 -5.0 -4.0 -3.0
4.0 3.860 3.846 3.828 3.8el 3.749 3.6363.0 3.855 3.840 3.822 3.7q5 3.744 3.6302.0 3.815 3 .. 800 3.782 3.755 3.703 3.5901.0 3.578 3.563 3.545 3.518 3.466 3.3510.0 2.744 2.729 2.711 2.684 2.631 2.512
-1 .. 0 1.470 1.455 1.437 1.. 4C9 1,,356 1.229
-2.0 0.607 0.592 0.574 0.546 0 .. 492 0.361-3 .. 0 0.248 0.234 0.216 0.188 0.133 0 .. 0-4.0 0.116 0.101 0.083 0.055 0 .. 0 0.0-5.0 0.061 0.046 0.028 0.0 0.0 0.0-6.0 0.032 0 .. 018 0.0 0.0 0.0 0.0-7.0 0.014 0.0 0.0 0.0 0.0 0.0
151
**cr~N=0.30 -Lc[ =6.00 **
** v =10 ~;31V ='1.40 **
L.No
LI\I1. -8.0 -7.0 -6.0 -5.0 -4.0 -3.0
4.0 4.059 3.878 3.728 3.574 3.375 3.0663~0 4.058 3.877 3.727 3.573 3.374 3.0652~0 4.048 3.867 3.718 3.563 3.364 3.0551 .. 0 3.963 3.782 3.632 3.478 3.279 2.969O~O 3.525 3.344 . 3. 1 94 3.040 2.840 2.528
-LO 2.556 2.375 2.226 2.071 1.870 1.553
-2 ~ 0 1.600 1.420 1.270 1.111) 0.913 0.591-3.0 1.013 0.832 0.683 0.527 0.325 0.0-4.0 0.688 0.507 0.358 0.202 0.0 0.0-5.0 0.486 0.305 0.15S 0.0 0.0 0.0-6GO 0.330 O. 150 C.O o. a 0.0 0.0-7.0 0.181 0.0 0.0 0.0 0.0 0.0
**CT~N =0.30 -LeI -=6.00 **
** v=10 /Aj3,y =1.80 **
LA/o
[,1/1.. -8.0 -7.0 -6.0 -5.0 -4.0 -3.0
4.0 7.694 5.750 4.677 3.937 3.298 2.6253.0 7.694 5.750 4.677 3.937 3.298 2.6252.0 7.692 5.748 4.674 3.935 3.296 2.6231.0 7.665 5.721 4.648 3.908 3.269 2.5960.0 7.467 5.523 4.450 3.110 3.071 2.396
-1.0 6.831 4.887 3.814 3.074 2.435 1.757
-2.0 5.917 3.973 2.900 2.160 1.519 0.836- 3.0 5.086 3.142 2.069 1.329 0.688 0.0-4.0 4.400 2.456 1.382 0.6142 0.0 0.0-5.0 3.758 1 • 814 0.740 0.0 0.0 0.0-6.0 3.017 1.073 c.o 0.0 0.0 0.0-7.0 1.944 0.0 0.0 0.0 0.0 0.0
